This section breaks down acceleration into two components called the tangential and normal components. Similar to how we break down all vectors into \(\hat{\textbf{i}}\), \( \hat{\textbf{j}} \), and \( \hat{\textbf{k}} \) components, we can do the same with acceleration. The addition of these two components will give us the overall acceleration.
We're use to thinking about acceleration as the second derivative of position, and while that is one way to look at the overall acceleration, we can further break down acceleration into two components: tangential and normal acceleration. The tangential acceleration, denoted \(a_T\)allows us to know how much of the acceleration acts in the direction of motion. The normal acceleration \(a_N\) is how much of the acceleration is orthogonal to the tangential acceleration.
Remember that vectors have magnitude AND direction. The tangential acceleration is a measure of the rate of change in the magnitude of the velocity vector, i.e. speed, and the normal acceleration are a measure of the rate of change of the direction of the velocity vector. This approach to acceleration is particularly useful in physics applications, because we need to know how much of the total acceleration acts in any given direction. Think for example of designing brakes for a car or the engine of a rocket. Why might it be useful to separate acceleration into components?
We can find the tangential accelration by using Chain Rule to rewrite the velocity vector as follows: \[\mathbf{v}=\dfrac{\mathrm{d\textbf{r}} }{\mathrm{d} t}=\dfrac{\mathrm{d\textbf{r}} }{\mathrm{d} s}\dfrac{\mathrm{d\textit{s}} }{\mathrm{d} t}=\textbf{T}\dfrac{\mathrm{d\mathit{s}} }{\mathrm{d} t}\] Now, since acceleration is simply the derivative of velocity, we find that: \[\begin{align} \mathbf{a}&=\dfrac{\mathrm{d\mathbf{v}} }{\mathrm{d} t}\\ &=\dfrac{\mathrm{d} }{\mathrm{d} t}(\mathbf{T}\dfrac{\mathrm{d\mathit{s}} }{\mathrm{d} t}) \\ &=\dfrac{\mathrm{d} ^2\mathit{s}}{\mathrm{d} t^2}\mathbf{T}+\dfrac{\mathrm{d} s}{\mathrm{d} t}\dfrac{\mathrm{d} \mathbf{T}}{\mathrm{d} t} \\ & =\dfrac{\mathrm{d} ^2s}{\mathrm{d} t^2}\mathbf{T}+\dfrac{\mathrm{d} s}{\mathrm{d} t}\left ( \dfrac{\mathrm{d} \mathbf{T}}{\mathrm{d} s}\dfrac{\mathrm{d} s}{\mathrm{d} t} \right ) \\ & =\dfrac{\mathrm{d} ^2s}{\mathrm{d} t^2}\mathbf{T}+\dfrac{\mathrm{d} s}{\mathrm{d} t}\left ( \kappa \mathbf{N}\dfrac{\mathrm{d} s}{\mathrm{d} t} \right ) \\ & =\dfrac{\mathrm{d} ^2s}{\mathrm{d} t^2}\mathbf{T}+\kappa \left ( \dfrac{\mathrm{d} s}{\mathrm{d} t} \right )^2\mathbf{N} \end{align}\]
Note \[\dfrac{\mathrm{d} \mathbf{T}}{\mathrm{d} s}=\kappa \mathbf{N}\] This, in turn, gives us the definition for acceleration by components. Definition: acceleration vector If the acceleration vector is written as \[\mathbf{a}=a_T\mathbf{T}+a_N\mathbf{N},\] then \[a_T=\dfrac{\mathrm{d} ^2s}{\mathrm{d} t^2}=\dfrac{\mathrm{d} }{\mathrm{d} t}|v| \ \text{ and } \ a_N=\kappa \left ( \dfrac{\mathrm{d} s}{\mathrm{d} t} \right )^2=\kappa|v|^2\] To calculate the normal component of the accleration, use the following formula: \[a_N=\sqrt{|a|^2-a_T^2} \label{Normal}\] We can relate this back to a common physics principal-uniform circular motion. In uniform circulation motion, when the speed is not changing, there is no tangential acceleration, only normal accleration pointing towards the center of circle. Why do you think this is? Hint: look in the introduction section for the difference between the two components of acceleration.
Example \(\PageIndex{1}\) Without finding T and N, write the accelration of the motion \[\mathbf{r}(t)=(\cos t+t\sin t) \hat{ \textbf{i} } +(\sin t-t\cos t)\hat{ \textbf{j} }\] for \(t>0\). To solve this problem, we must first find the particle's velocity. \[\begin{align} \mathbf{v}&=\dfrac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} \\ & =(-\sin t+\sin t+t\cos t) \hat{ \textbf{i} } +(\cos t-\cos t+t\sin t) \hat{ \textbf{j} } \\ &=(t\cos t) \hat{ \textbf{i} } +(t\sin t) \hat{ \textbf{j} } \end{align}\] Next find the speed. \[|v|=\sqrt{t^2\cos ^2t+t^2\sin ^2t}=\sqrt{t^2}=|t|\] When \(t>0\), \(|t|\) simply becomes \(t\). We know that \(a_T=\dfrac{\mathrm{d} }{\mathrm{d} t}|v|\), which we can use to find that \(\dfrac{\mathrm{d} }{\mathrm{d} t}(t)=1\). Now that we know \(a_T\), we can use it to find \(a_N\) using Equation \(\ref{Normal}\). \[\mathbf{a}=(\cos t-t\sin t) \hat{ \textbf{i} } +(\sin t+t\cos t) \hat{ \textbf{j} } \] \[|\mathbf{a}|^2=t^2+1\] \[a_N=\sqrt{(t^2+1)-(1)}=t\] By substituting this back into the original definition, we find that \[|\mathbf{a}|=(1)\mathbf{T}+(t)\mathbf{N}=\mathbf{T}+t\mathbf{N}\]
Example \(\PageIndex{2}\) Write \(a\)in the form \(\mathbf{a}=a_T\mathbf{T}+a_N\mathbf{N}\) without finding T or N. \[\mathbf{r}(t)=(t+1) \hat{ \textbf{i} } +2t \hat{ \textbf{j} } +t^2 \hat{ \textbf{k} } \] \[\mathbf{v}=t \hat{ \textbf{i} } +2 \hat{\textbf{j}} +2t \hat{ \textbf{k} } \] \[|\mathbf{v}|=\sqrt{5+4t^2}\] \[a_T=\dfrac{1}{2}\left ( 5+4t^2 \right )^{-\dfrac{1}{2}}(8t)\] \[=4t(5+4t^2)^{-\dfrac{1}{2}}\] \[a_T(1)=\dfrac{4}{\sqrt{9}}=\dfrac{4}{3}\] \[\mathbf{a}=2 \hat{ \textbf{k} } \] \[\mathbf{a}(1)=2 \hat{ \textbf{k} } \] \[\mathbf{a}(t)=2 \hat{ \textbf{k} } \] Now we use Equation \(\ref{Normal}\): \[\begin{align} a_N=\sqrt{|\mathbf{a}|^2-a_T^2}&=\sqrt{2^2-\left ( \dfrac{4}{3} \right )^2} \\ & =\sqrt{\dfrac{20}{9}} \\ &=\dfrac{2\sqrt{5}}{3} \end{align}\] \[\mathbf{a}(1)=\dfrac{4}{3}\mathbf{T}+\dfrac{2\sqrt{5}}{3}\mathbf{N}\]
We have now seen how to describe curves in the plane and in space, and how to determine their properties, such as arc length and curvature. All of this leads to the main goal of this chapter, which is the description of motion along plane curves and space curves. We now have all the tools we need; in this section, we put these ideas together and look at how to use them.
Our starting point is using vector-valued functions to represent the position of an object as a function of time. All of the following material can be applied either to curves in the plane or to space curves. For example, when we look at the orbit of the planets, the curves defining these orbits all lie in a plane because they are elliptical. However, a particle traveling along a helix moves on a curve in three dimensions.
Let \(\vecs r(t)\) be a twice-differentiable vector-valued function of the parameter \(t\) that represents the position of an object as a function of time. The velocity vector \(\vecs v(t)\) of the object is given by \[\text{Velocity}\,=\vecs v(t)=\vecs r′(t). \label{Eq1} \] The acceleration vector \(\vecs a(t)\) is defined to be \[\text{Acceleration}\,=\vecs a(t)=\vecs v′(t)=\vecs r″(t). \label{Eq2} \] The speed is defined to be \[\mathrm{Speed}\,=v(t)=‖\vecs v(t)‖=‖\vecs r′(t)‖=\dfrac{ds}{dt}. \label{Eq3} \] Since \(\vecs{r}(t)\) can be in either two or three dimensions, these vector-valued functions can have either two or three components. In two dimensions, we define \(\vecs{r}(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}\) and in three dimensions \(\vecs r(t)=x(t) \hat{\mathbf i}+y(t) \hat{\mathbf j}+z(t) \hat{\mathbf k}\). Then the velocity, acceleration, and speed can be written as shown in the following table.
A particle moves in a parabolic path defined by the vector-valued function \(\vecs{r}(t)=t^2 \hat{\mathbf i}+ \sqrt{5−t^2} \hat{\mathbf j}\), where \(t\) measures time in seconds.
Solution
A particle moves in a path defined by the vector-valued function \(\vecs r(t)=(t^2−3t)\,\hat{\mathbf i}+(2t−4)\,\hat{\mathbf j}+(t+2)\,\hat{\mathbf k}\), where \(t\) measures time in seconds and where distance is measured in feet. Find the velocity, acceleration, and speed as functions of time. HintUse Equations \ref{Eq1}, \ref{Eq2}, and \ref{Eq3}. Answer\[\begin{align*}\vecs v(t) &=\vecs{r}'(t) =(2t-3)\,\hat{\mathbf i}+2\,\hat{\mathbf j}+\,\hat{\mathbf k}\\[4pt] \vecs a(t) &=\vecs v′(t) =2\,\hat{\mathbf i} \end{align*}\] \[ ||\vecs{r}′(t)||=\sqrt{(2t-3)^2+2^2+1^2} =\sqrt{4t^2-12t+14} \nonumber \] The units for velocity and speed are feet per second, and the units for acceleration are feet per second squared. To gain a better understanding of the velocity and acceleration vectors, imagine you are driving along a curvy road. If you do not turn the steering wheel, you would continue in a straight line and run off the road. The speed at which you are traveling when you run off the road, coupled with the direction, gives a vector representing your velocity, as illustrated in Figure \(\PageIndex{2}\). Figure \(\PageIndex{2}\): At each point along a road traveled by a car, the velocity vector of the car is tangent to the path traveled by the car.However, the fact that you must turn the steering wheel to stay on the road indicates that your velocity is always changing (even if your speed is not) because your direction is constantly changing to keep you on the road. As you turn to the right, your acceleration vector also points to the right. As you turn to the left, your acceleration vector points to the left. This indicates that your velocity and acceleration vectors are constantly changing, regardless of whether your actual speed varies (Figure \(\PageIndex{3}\)). Figure \(\PageIndex{3}\): The dashed line represents the trajectory of an object (a car, for example). The acceleration vector points toward the inside of the turn at all times.
We can combine some of the concepts discussed in Arc Length and Curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. Recall that the unit tangent vector \(\vecs T\) and the unit normal vector \(\vecs N\) form an osculating plane at any point \(P\) on the curve defined by a vector-valued function \(\vecs{r}(t)\). The following theorem shows that the acceleration vector \(\vecs{a}(t)\) lies in the osculating plane and can be written as a linear combination of the unit tangent and the unit normal vectors.
The acceleration vector \(\vecs{a}(t)\) of an object moving along a curve traced out by a twice-differentiable function \(\vecs{r}(t)\) lies in the plane formed by the unit tangent vector \(\vecs T(t)\) and the principal unit normal vector \(\vecs N(t)\) to \(C\). Furthermore, \[\vecs{a}(t) = v'(t)\vecs{T}(t) + [v(t)]^2 \kappa \vecs{N}(t) \nonumber \] Here, \(v(t) = \|\vecs v(t)\|\) is the speed of the object and \(\kappa\) is the curvature of \(C\) traced out by \(\vecs{r}(t)\).
Because \(\vecs{v}(t)=\vecs{r}′(t)\) and \(\vecs{T}(t)=\dfrac{\vecs{r}′(t)}{||\vecs{r}′(t)||}\), we have \(\vecs v(t)=||\vecs{r}′(t)||\vecs{T}(t)=v(t)\vecs{T}(t)\). Now we differentiate this equation: \[\vecs{a}(t)=\vecs{v}′(t)=\dfrac{d}{dt}\left(v(t)\vecs{T}(t)\right)=v′(t)\vecs{T}(t)+v(t)\vecs{T}′(t) \nonumber \] Since \(\vecs{N}(t)=\dfrac{\vecs{T}′(t)}{||\vecs{T}′(t)||}\), we know \(\vecs{T}′(t)=||\vecs{T}′(t)||\vecs{N}(t)\), so \[\vecs{a}(t)=v′(t)\vecs{T}(t)+v(t)||\vecs{T}′(t)||\vecs{N}(t). \nonumber \] A formula for curvature is \(\kappa=\dfrac{||\vecs{T}'(t)||}{||\vecs{r}'(t)||}\), so \(\vecs{T}'(t) = \kappa ||\vecs{r}'(t) || = \kappa v(t) \). This gives \(\vecs{a}(t)=v′(t)\vecs{T}(t)+\kappa (v(t))^2 \vecs{N}(t).\) \(\square\) The coefficients of \(\vecs{T}(t)\) and \(\vecs{N}(t)\) are referred to as the tangential component of acceleration and the normal component of acceleration, respectively. We write \(a_\vecs{T}\) to denote the tangential component and \(a_\vecs{N}\) to denote the normal component.
Let \(\vecs{r}(t)\) be a vector-valued function that denotes the position of an object as a function of time. Then \(\vecs{a}(t)=\vecs{r}′′(t)\) is the acceleration vector. The tangential and normal components of acceleration \(a_\vecs{T}\) and \(a_\vecs{N}\) are given by the formulas \[a_{\vecs{T}}=\vecs a \cdot\vecs{T}=\dfrac{\vecs{v}\cdot\vecs{a}}{||\vecs{v}||} \label{Eq1B} \] and \[a_\vecs{N}=\vecs a\cdot \vecs N=\dfrac{||\vecs v \times \vecs a||}{||\vecs v||}=\sqrt{||\vecs a||^2−{\left(a_{\vecs{T}}\right)^2}}. \label{Eq2B} \] These components are related by the formula \[\vecs{a}(t)=a_\vecs{T} \vecs{T}(t)+a_\vecs{N}\vecs{N}(t). \label{Eq3B} \] Here \(\vecs{T}(t)\) is the unit tangent vector to the curve defined by \(\vecs{r}(t)\), and \(\vecs{N}(t)\) is the unit normal vector to the curve defined by \(\vecs{r}(t)\). The normal component of acceleration is also called the centripetal component of acceleration or sometimes the radial component of acceleration. To understand centripetal acceleration, suppose you are traveling in a car on a circular track at a constant speed. Then, as we saw earlier, the acceleration vector points toward the center of the track at all times. As a rider in the car, you feel a pull toward the outside of the track because you are constantly turning. This sensation acts in the opposite direction of centripetal acceleration. The same holds true for non-circular paths. The reason is that your body tends to travel in a straight line and resists the force resulting from acceleration that push it toward the side. Note that at point \(B\) in Figure \(\PageIndex{4}\) the acceleration vector is pointing backward. This is because the car is decelerating as it goes into the curve. Figure \(\PageIndex{4}\): The tangential and normal components of acceleration can be used to describe the acceleration vector.The tangential and normal unit vectors at any given point on the curve provide a frame of reference at that point. The tangential and normal components of acceleration are the projections of the acceleration vector onto \(\vecs T\) and \(\vecs N\), respectively.
A particle moves in a path defined by the vector-valued function \(\vecs{r}(t)=t^2\,\hat{\mathbf i}+(2t−3)\,\hat{\mathbf j}+(3t^2−3t)\,\hat{\mathbf k}\), where \(t\) measures time in seconds and distance is measured in feet.
Solution
An object moves in a path defined by the vector-valued function \(\vecs r(t)=4t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}\), where \(t\) measures time in seconds.
Use Equations \ref{Eq1B} and \ref{Eq2B} Answera. \[\begin{align*} a_\vecs{T} =\dfrac{\vecs v(t) \cdot \vecs a(t)}{||\vecs v(t)||}= \dfrac{\vecs r'(t) \cdot \vecs r''(t) }{||\vecs r'(t)||} \\ = \dfrac{ (4\,\hat{\mathbf i} + 2t \,\hat{\mathbf j}) \cdot (2\,\hat{\mathbf j}) }{||4\,\hat{\mathbf i} + 2t \,\hat{\mathbf j} ||} \\ = \dfrac{4t}{\sqrt{4^2 + (2t)^2}}\\ = \dfrac{2t}{\sqrt{2+t^2}} \end{align*}\] b. \[\begin{align*} a_\vecs{T}(−3) = \dfrac{2(-3)}{\sqrt{2+(-3)^2}} \\ = \dfrac{-6}{\sqrt{11}}\end{align*}\]
Now let’s look at an application of vector functions. In particular, let’s consider the effect of gravity on the motion of an object as it travels through the air, and how it determines the resulting trajectory of that object. In the following, we ignore the effect of air resistance. This situation, with an object moving with an initial velocity but with no forces acting on it other than gravity, is known as projectile motion. It describes the motion of objects from golf balls to baseballs, and from arrows to cannonballs. First we need to choose a coordinate system. If we are standing at the origin of this coordinate system, then we choose the positive \(y\)-axis to be up, the negative \(y\)-axis to be down, and the positive \(x\)-axis to be forward (i.e., away from the thrower of the object). The effect of gravity is in a downward direction, so Newton’s second law tells us that the force on the object resulting from gravity is equal to the mass of the object times the acceleration resulting from gravity, or \(\vecs F_g=m\vecs a\), where \(\vecs F_g\) represents the force from gravity and \(\vecs a = -g\,\hat{\mathbf j}\) represents the acceleration resulting from gravity at Earth’s surface. The value of \(g\) in the English system of measurement is approximately 32 ft/sec2 and it is approximately 9.8 m/sec2 in the metric system. This is the only force acting on the object. Since gravity acts in a downward direction, we can write the force resulting from gravity in the form \(\vecs F_g=−mg\,\hat{\mathbf j}\), as shown in Figure \(\PageIndex{5}\). Figure \(\PageIndex{5}\): An object is falling under the influence of gravity.Newton’s second law also tells us that \(F=m\vecs{a}\), where \(\vecs a\) represents the acceleration vector of the object. This force must be equal to the force of gravity at all times, so we therefore know that \[\begin{align*} \vecs F =\vecs F_g \\ m\vecs{a} = -mg \,\hat{\mathbf j} \\ \vecs{a} = -g\,\hat{\mathbf j}. \end{align*}\] Now we use the fact that the acceleration vector is the first derivative of the velocity vector. Therefore, we can rewrite the last equation in the form \[\vecs v'(t) = -g\,\hat{\mathbf j} \nonumber \] By taking the antiderivative of each side of this equation we obtain \[ \vecs v(t) = \int -g \,\hat{\mathbf j}\; dt = -gt\,\hat{\mathbf j} + \vecs C_1 \nonumber \] for some constant vector \(\vecs C_1\). To determine the value of this vector, we can use the velocity of the object at a fixed time, say at time \(t=0\). We call this velocity the initial velocity: \(\vecs v(0)=\vecs v_0\). Therefore, \(\vecs v(0)=−g(0)\,\hat{\mathbf j}+\vecs C_1=\vecs v_0\) and \(\vecs C_1= \vecs v_0\). This gives the velocity vector as \(\vecs v(t)=−gt\,\hat{\mathbf j}+\vecs v_0\). Next we use the fact that velocity \(\vecs{v}(t)\) is the derivative of position \(\vecs{s}(t)\). This gives the equation \[\vecs s'(t)=-gt\,\hat{\mathbf j}+\vecs{v}_0. \nonumber \] Taking the antiderivative of both sides of this equation leads to \[\begin{align*} \vecs s(t) &= \int -gt\,\hat{\mathbf j} + \vecs{v}_0 \;dt \\[4pt] &= -\dfrac{1}{2}gt^2 \,\hat{\mathbf j} + \vecs{v}_0 t + \vecs{C}_2 \end{align*}\] with another unknown constant vector \(\vecs{C}_2\). To determine the value of \(\vecs{C}_2\), we can use the position of the object at a given time, say at time \(t=0\). We call this position the initial position: \(\vecs{s}(0)=\vecs{s}_0\). Therefore, \(\vecs{s}(0)=−(1/2)g(0)^2\,\hat{\mathbf j}+\vecs{v}_0(0)+\vecs{C}_2=\vecs{s}_0\). This gives the position of the object at any time as \[ \vecs{s}(t)=−\dfrac{1}{2}gt^2 \,\hat{\mathbf j}+\vecs{v}_0 t+\vecs{s}_0. \nonumber \] Let’s take a closer look at the initial velocity and initial position. In particular, suppose the object is thrown upward from the origin at an angle \(\theta\) to the horizontal, with initial speed \(\vecs{v}_0\). How can we modify the previous result to reflect this scenario? First, we can assume it is thrown from the origin. If not, then we can move the origin to the point from where it is thrown. Therefore, \(\vecs{s}_0=\vecs{0}\), as shown in Figure \(\PageIndex{6}\). Figure \(\PageIndex{6}\): Projectile motion when the object is thrown upward at an angle θ. The horizontal motion is at constant velocity and the vertical motion is at constant acceleration.We can rewrite the initial velocity vector in the form \(\vecs{v}_0= v_0 \cos \theta \,\hat{\mathbf i} + v_0 \sin \theta \,\hat{\mathbf j}\). Then the equation for the position function \(\vecs{s}(t)\) becomes \[\begin{align*} \vecs{s}(t) &=-\dfrac{1}{2} gt^2\,\hat{\mathbf j} + v_0 t \cos\theta \,\hat{\mathbf i} + v_0 t \sin\theta \,\hat{\mathbf j} \\[4pt] &= v_0 t \cos\theta\,\hat{\mathbf i} + v_0 t \sin\theta \,\hat{\mathbf j} - \dfrac{1}{2} gt^2\,\hat{\mathbf j} \\[4pt] &= v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2} gt^2\right)\,\hat{\mathbf j}. \end{align*}\] The coefficient of \(\hat{\mathbf i}\) represents the horizontal component of \(\vecs{s}(t)\) and is the horizontal distance of the object from the origin at time \(t\). The maximum value of the horizontal distance (measured at the same initial and final altitude) is called the range \(R\). The coefficient of \(\hat{\mathbf j}\) represents the vertical component of \(\vecs{s}(t)\) and is the altitude of the object at time \(t\). The maximum value of the vertical distance is the height \(H\).
During an Independence Day celebration, a cannonball is fired from a cannon on a cliff toward the water. The cannon is aimed at an angle of 30° above horizontal and the initial speed of the cannonball is 600 ft/sec. The cliff is 100 ft above the water (Figure \(\PageIndex{7}\)).
Solution We use the equation \[\vecs{s}(t) = v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0 t \sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j} \nonumber \] with \(\theta=30^\circ \), \(g=32 \dfrac{\text{ft}}{\text{sec}^2}\), and \(v_0=600 \dfrac{\text{ft}}{\text{sec}^2}\). Then the position equation becomes \[\begin{align*} \vecs{s}(t) &= 600 t ( \cos 30^\circ)\,\hat{\mathbf i} + \left(600t \sin30^\circ - \dfrac{1}{2}(32)t^2 \right)\,\hat{\mathbf j} \\[4pt] &= 300t\sqrt{3} \,\hat{\mathbf i} + \left( 300t - 16t^2 \right)\,\hat{\mathbf j} \end{align*}\]
An archer fires an arrow at an angle of 40° above the horizontal with an initial speed of 98 m/sec. The height of the archer is 171.5 cm. Find the horizontal distance the arrow travels before it hits the ground. HintThe equation for the position vector needs to account for the height of the archer in meters. Answer967.15 m One final question remains: In general, what is the maximum distance a projectile can travel, given its initial speed? To determine this distance, we assume the projectile is fired from ground level and we wish it to return to ground level. In other words, we want to determine an equation for the range. In this case, the equation of projectile motion is \[\vecs{s}=v_0 t \cos\theta \,\hat{\mathbf i} + \left(v_0t\sin\theta - \dfrac{1}{2}gt^2 \right)\,\hat{\mathbf j}. \nonumber \] Setting the second component equal to zero and solving for \(t\) yields \[\begin{align*} v_0 t \sin\theta - \dfrac{1}{2}gt^2 =0\\ t\left(v_0 \sin\theta - \dfrac{1}{2}gt\right) =0 \end{align*}\] Therefore, either \(t=0\) or \(t=\dfrac{2v_0\sin\theta}{g}\). We are interested in the second value of \(t\), so we substitute this into \(\vecs{s}(t)\), which gives \[\begin{align*} \vecs{s}\left(\dfrac{2v_0\sin\theta}{g} \right) = v_0 \left(\dfrac{2v_0\sin\theta}{g} \right) \cos\theta \,\hat{\mathbf i} + \left( v_0\left(\dfrac{2v_0\sin\theta}{g} \right)\sin\theta - \dfrac{1}{2}g\left(\dfrac{2v_0\sin\theta}{g} \right)^2 \right)\,\hat{\mathbf j} \\ = \left(\dfrac{2v_0^2\sin\theta\cos\theta}{g} \right)\,\hat{\mathbf i} \\ = \dfrac{v_0^2 \sin2\theta}{g}\,\hat{\mathbf i}. \end{align*}\] Thus, the expression for the range of a projectile fired at an angle \(\theta\) is \[R=\dfrac{v_0^2 \sin2\theta}{g}\,\hat{\mathbf i} . \nonumber \] The only variable in this expression is \( \theta\). To maximize the distance traveled, take the derivative of the coefficient of i with respect to \(\theta\) and set it equal to zero: \[\begin{align*} \dfrac{d}{d\theta} \left( \dfrac{v_0^2 \sin2\theta}{g} \right) =0\\ \dfrac{2v_0^2\cos2\theta}{g} =0\\ \theta=45^\circ \end{align*}\] This value of \(\theta)\) is the smallest positive value that makes the derivative equal to zero. Therefore, in the absence of air resistance, the best angle to fire a projectile (to maximize the range) is at a 45° angle. The distance it travels is given by \[\vecs{s}\left(\dfrac{2v_0 \sin 45^\circ}{g} \right)= \dfrac{v_0^2 \sin 90^\circ}{g} \,\hat{\mathbf i} = \dfrac{v_0^2}{g}\,\hat{\mathbf i} \nonumber \] Therefore, the range for an angle of 45° is \(\frac{v_0^2}{g}\) units.
During the early 1600s, Johannes Kepler was able to use the amazingly accurate data from his mentor Tycho Brahe to formulate his three laws of planetary motion, now known as Kepler’s laws of planetary motion. These laws also apply to other objects in the solar system in orbit around the Sun, such as comets (e.g., Halley’s comet) and asteroids. Variations of these laws apply to satellites in orbit around Earth.
Kepler’s third law is especially useful when using appropriate units. In particular, 1 astronomical unit is defined to be the average distance from Earth to the Sun, and is now recognized to be 149,597,870,700 m or, approximately 93,000,000 mi. We therefore write 1 A.U. = 93,000,000 mi. Since the time it takes for Earth to orbit the Sun is 1 year, we use Earth years for units of time. Then, substituting 1 year for the period of Earth and 1 A.U. for the average distance to the Sun, Kepler’s third law can be written as \[ T_p^2=D_p^3 \nonumber \] for any planet in the solar system, where \(T_P\) is the period of that planet measured in Earth years and \(D_P\) is the average distance from that planet to the Sun measured in astronomical units. Therefore, if we know the average distance from a planet to the Sun (in astronomical units), we can then calculate the length of its year (in Earth years), and vice versa. Kepler’s laws were formulated based on observations from Brahe; however, they were not proved formally until Sir Isaac Newton was able to apply calculus. Furthermore, Newton was able to generalize Kepler’s third law to other orbital systems, such as a moon orbiting around a planet. Kepler’s original third law only applies to objects orbiting the Sun.
Let’s now prove Kepler’s first law using the calculus of vector-valued functions. First we need a coordinate system. Let’s place the Sun at the origin of the coordinate system and let the vector-valued function \(\vecs{r}(t)\) represent the location of a planet as a function of time. Newton proved Kepler’s law using his second law of motion and his law of universal gravitation. Newton’s second law of motion can be written as \(\vecs{F}=m\vecs{a}\), where \(\vecs{F}\) represents the net force acting on the planet. His law of universal gravitation can be written in the form \(\vecs{F}=−\dfrac{GmM}{||\vecs{r}||^2}\cdot \dfrac{\vecs{r}}{||\vecs{r} ||}\), which indicates that the force resulting from the gravitational attraction of the Sun points back toward the Sun, and has magnitude \(\dfrac{GmM}{||\vecs{r}||^2} \) (Figure \(\PageIndex{9}\)). Figure \(\PageIndex{9}\): The gravitational force between Earth and the Sun is equal to the mass of the earth times its acceleration.Setting these two forces equal to each other, and using the fact that \(\vecs a(t)=\vecs v′(t)\), we obtain \[ m\vecs v′(t)=−\frac{GmM}{‖\vecs r‖^2}⋅\frac{\vecs r}{‖\vecs r‖}, \nonumber \] which can be rewritten as \[ \dfrac{d\vecs v}{dt}=−\dfrac{GM}{||\vecs r||^3}\vecs{r}. \nonumber \] This equation shows that the vectors \(d\vecs{v}/dt\) and \(\vecs r\) are parallel to each other, so \(d\vecs {v}/dt \times \vecs {r}=\vecs 0\). Next, let’s differentiate \(\vecs{r} \times \vecs{v}\) with respect to time: \[\dfrac{d}{dt}(\vecs{r}\times \vecs{v})=\dfrac{d\vecs{r}}{dt}\times \vecs v+\vecs{r} \times \dfrac{d\vecs{v}}{dt}=\vecs{v}\times \vecs{v}+\vecs{0}=\vecs{0}. \label{Eq10} \] This proves that \(\vecs{r}\times\vecs{v}\) is a constant vector, which we call \(\vecs C\). Since \(\vecs r\) and \(\vecs v\) are both perpendicular to \(\vecs C\) for all values of \(t\), they must lie in a plane perpendicular to \(\vecs C\). Therefore, the motion of the planet lies in a plane. Next we calculate the expression \(d\vecs{v}/dt\times \vecs C\): \[\dfrac{d\vecs{v}}{dt} \times \vecs{C}=−\dfrac{GM}{||\vecs{r}||^3}\vecs{r}\times (\vecs{r}\times\vecs{v})=−\dfrac{GM}{||\vecs r||^3}[(\vecs{r} \cdot \vecs{v})\vecs{r} - (\vecs{r} \cdot \vecs{r})\vecs{v}]. \label{Eq11} \] The last equality in Equation \ref{Eq10} is from the triple cross product formula (Introduction to Vectors in Space). We need an expression for \(\vecs{r}\cdot \vecs{v}\). To calculate this, we differentiate \(\vecs{r}\cdot \vecs{r}\) with respect to time: \[ \dfrac{d}{dt}(\vecs{r}\cdot \vecs{r})=\dfrac{d\vecs{r}}{dt}\cdot \vecs{r}+\vecs{r}\cdot \dfrac{d\vecs{r}}{dt}=2\vecs{r}\cdot \dfrac{d\vecs{r}}{dt}=2\vecs{r}\cdot \vecs{v}. \label{Eq12} \] Since \(\vecs{r}\cdot\vecs{r}=||\vecs r||^2\), we also have \[\dfrac{d}{dt}(\vecs{r}\cdot \vecs{r})=\dfrac{d}{dt}||\vecs{r}||^2=2||\vecs{r}|| \dfrac{d}{dt}||\vecs{r}||. \label{Eq13} \] Combining Equation \ref{Eq12} and Equation \ref{Eq13}, we get \[\begin{align*} 2\vecs{r}\cdot \vecs{v} =2||\vecs{r}||\dfrac{d}{dt}||\vecs{r}|| \\ \vecs{r} \cdot \vecs{v} =||\vecs{r}‖\dfrac{d}{dt}||\vecs{r}||. \end{align*} \label{Eq14} \] Substituting this into Equation \ref{Eq11} gives us \[\begin{align} \dfrac{d\vecs{v}}{dt} \times \vecs{C} = - \dfrac{GM}{||\vecs{r}||^3} [(\vecs{r}\cdot \vecs{v})\vecs{r} - (\vecs{r}\cdot \vecs{r})\vecs{v}] \nonumber \\ = -\dfrac{GM}{||\vecs{r}||^3}\left[ ||\vecs{r} \left(\dfrac{d}{dt} ||\vecs{r}||\right)\vecs{r} - ||\vecs{r}||^2\vecs{v} \right] \nonumber \\ = -GM\left[ \dfrac{1}{||\vecs{r}||^2}\left( \dfrac{d}{dt} ||\vecs{r}|| \right)\vecs{r} - \dfrac{1}{||\vecs{r}||}\vecs{v} \right] \nonumber \\ = GM\left[ \dfrac{\vecs{v}}{||\vecs{r}||} -\dfrac{\vecs{r}}{||\vecs{r}||^2}\left( \dfrac{d}{dt} ||\vecs{r}|| \right) \right]. \label{Eq15} \end{align} \] However, \[ \begin{align*} \dfrac{d}{dt} \dfrac{\vecs{r}}{||\vecs{r}||} = \dfrac{ \frac{d}{dt}(\vecs{r})||\vecs{r}||- \vecs{r}\frac{d}{dt}||\vecs{r}|| }{||\vecs{r}||^2} \\ = \dfrac{ \frac{d\vecs{r}}{dt} }{||\vecs{r}||} - \dfrac{\vecs{r}}{||\vecs{r}||^2}\dfrac{d}{dt}||\vecs{r} || \\ = \dfrac{\vecs{v}}{||\vecs{r}||} - \dfrac{\vecs{r}}{||\vecs{r}||^2} \dfrac{d}{dt}||\vecs{r}||. \end{align*}\] Therefore, Equation \ref{Eq15} becomes \[\dfrac{d \vecs{v}}{dt}\times \vecs{C}=GM\left( \dfrac{d}{dt}\dfrac{ \vecs{r}}{ || \vecs{r} ||} \right).\nonumber \] Since \(\vecs{C}\) is a constant vector, we can integrate both sides and obtain \[ \vecs{v}\times\vecs{C} = GM \dfrac{ \vecs{r} }{|| \vecs{r} ||} + \vecs{D}, \nonumber \] where \(\vecs D\) is a constant vector. Our goal is to solve for \(|| \vecs{r} ||\). Let’s start by calculating \( \vecs{r} \cdot ( \vecs{v}\times \vecs{C}\): \[\vecs{r} \cdot ( \vecs{v}\times \vecs{C} =GM\dfrac{||\vecs{r}||^2}{||\vecs{r}||}+ \vecs{r}\cdot\vecs{D} =GM||\vecs{r}||+\vecs{r}\cdot \vecs{D}. \nonumber \] However, \( \vecs{r} \cdot ( \vecs{v}\times \vecs{C})= ( \vecs{r} \times \vecs{v})\cdot \vecs{C} \), so \[ ( \vecs{r} \times \vecs{v})\cdot \vecs{C} =GM||\vecs{r}|| + \vecs{r}\cdot \vecs{D}.\nonumber \] Since \(\vecs{r}\times \vecs{v}=\vecs{C}\), we have \[ ||\vecs{C}||^2 =GM||\vecs{r}|| +\vecs{r}\cdot \vecs{D}.\nonumber \] Note that \( \vecs{r} \cdot \vecs{D}=||\vecs{r}|| ||\vecs{D}||\cos \theta \), where \(\theta\) is the angle between \(\vecs{r}\) and \(\vecs{D}\). Therefore, \[ ||\vecs{C}||^2=GM||\vecs{r}||+||\vecs{r}|| ||\vecs{D}|| \cos\theta \nonumber \] Solving for \(||\vecs{r}||\), \[ ||\vecs{r}|| = \dfrac{||\vecs{C}||^2 }{GM+||\vecs{D}||\cos\theta} = \dfrac{||\vecs{C}||^2}{GM}\left( \dfrac{1}{1+e\cos\theta} \right). \nonumber \] where \(e=||\vecs{D}||/GM\). This is the polar equation of a conic with a focus at the origin, which we set up to be the Sun. It is a hyperbola if \(e>1\), a parabola if \(e=1\), or an ellipse if \(e<1\). Since planets have closed orbits, the only possibility is an ellipse. However, at this point it should be mentioned that hyperbolic comets do exist. These are objects that are merely passing through the solar system at speeds too great to be trapped into orbit around the Sun. As they pass close enough to the Sun, the gravitational field of the Sun deflects the trajectory enough so the path becomes hyperbolic. \(\square\) Kepler’s third law of planetary motion can be modified to the case of one object in orbit around an object other than the Sun, such as the Moon around the Earth. In this case, Kepler’s third law becomes \[P^2 = \dfrac{4\pi^2 a^3}{G(m+M)}, \label{Eq30} \] where m is the mass of the Moon and M is the mass of Earth, a represents the length of the major axis of the elliptical orbit, and P represents the period.
Given that the mass of the Moon is \(7.35\times 10^{22}\) kg, the mass of Earth is \(5.97\times 10^{24}\) kg, \(G=6.67\times 10^{−11} \text{m} / \text{kg} \cdot \text{sec}^2\), and the period of the moon is 27.3 days, let’s find the length of the major axis of the orbit of the Moon around Earth. Solution It is important to be consistent with units. Since the universal gravitational constant contains seconds in the units, we need to use seconds for the period of the Moon as well: \[27.3 \; \text{days} \times \dfrac{24 \; \text{hr}}{1 \; \text{day}} \times \dfrac{3600 \; \text{esc}}{1 \; \text{hour}} =2,358,720\; \text{sec}\nonumber \] Substitute all the data into Equation \ref{Eq30} and solve for \(a\): \[\begin{align*} (2,358,720sec)^2 = \dfrac{4\pi^2a^3}{\left( 6.67\times 10^{-11} \frac{m}{\text{kg}\times \text{sec}^2}\right) (7.35\times 10^{22}\text{kg} + 5.97 \times 10^{24}\text{kg})} \\ 5.563 \times 10^{12} = \dfrac{ 4\pi^2a^3}{(6.67 \times 10^{-11}\text{m}^3)(6.04 \times 10^{24})} \\ (5.563 \times 10^{12})(6.67 \times 10^{-11} \text{m}^3)(6.04 \times 10^{24}) = 4\pi^2 a^3 \\ a^3 = \dfrac{2.241 \times 10^{27}}{4\pi^2}\text{m}^3 \\ a = 3.84 \times 10^8 \text{m} \\ \approx 384,000 \,\text{km}. \end{align*}\] Analysis According to solarsystem.nasa.gov, the actual average distance from the Moon to Earth is 384,400 km. This is calculated using reflectors left on the Moon by Apollo astronauts back in the 1960s.
Titan is the largest moon of Saturn. The mass of Titan is approximately \(1.35 \times 10^{23} kg\). The mass of Saturn is approximately \( 5.68 \times 10^{26}\) kg. Titan takes approximately 16 days to orbit Saturn. Use this information, along with the universal gravitation constant \(G=6.67×10^{−11} \text{m}/\text{kg} \cdot \text{sec}^2\) to estimate the distance from Titan to Saturn. HintMake sure your units agree, then use Equation \ref{Eq30}. Answer\[a\approx 1.224 \times 10^9 \text{m}= 1,224,000 \text{km} \nonumber \]
We now return to the chapter opener, which discusses the motion of Halley’s comet around the Sun. Kepler’s first law states that Halley’s comet follows an elliptical path around the Sun, with the Sun as one focus of the ellipse. The period of Halley’s comet is approximately 76.1 years, depending on how closely it passes by Jupiter and Saturn as it passes through the outer solar system. Let’s use \(T=76.1\) years. What is the average distance of Halley’s comet from the Sun? Solution Using the equation \(T^2=D^3\) with \(T=76.1\), we obtain \(D^3=5791.21\), so \(D\approx 17.96\) A.U. This comes out to approximately \(1.67\times 10^9\) mi. A natural question to ask is: What are the maximum (aphelion) and minimum (perihelion) distances from Halley’s Comet to the Sun? The eccentricity of the orbit of Halley’s Comet is 0.967 (Source: http://nssdc.gsfc.nasa.gov/planetary...cometfact.html). Recall that the formula for the eccentricity of an ellipse is \(e=c/a\), where a is the length of the semimajor axis and c is the distance from the center to either focus. Therefore, \(0.967=c/17.96\) and \(c\approx 17.37\) A.U. Subtracting this from a gives the perihelion distance \(p=a−c=17.96−17.37=0.59\) A.U. According to the National Space Science Data Center (Source: http://nssdc.gsfc.nasa.gov/planetary...cometfact.html), the perihelion distance for Halley’s comet is 0.587 A.U. To calculate the aphelion distance, we add \[ P=a+c=17.96+17.37=35.33 \; \text{A.U.} \nonumber \] This is approximately \(3.3\times 10^9\) mi. The average distance from Pluto to the Sun is 39.5 A.U. (Source: http://www.oarval.org/furthest.htm), so it would appear that Halley’s Comet stays just within the orbit of Pluto.
How fast can a racecar travel through a circular turn without skidding and hitting the wall? The answer could depend on several factors:
In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass \(m\) moves with constant angular speed \(\omega\) around a circular curve of radius \(R\) (Figure \(\PageIndex{9}\)). The curve is banked at an angle \(\theta\). If the height of the car off the ground is \(h\), then the position of the car at time \(t\) is given by the function \(\vecs r(t)=< R\cos(\omega t),R\sin(\omega t),h>\). Figure \(\PageIndex{9}\): Views of a race car moving around a track.
As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure \(\PageIndex{10}\)). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that \(\vecs{f}=\mu \vecs{N}\) for some positive constant \(\mu \). The constant \(\mu\) is called the coefficient of friction. Figure \(\PageIndex{10}\): The car has three forces acting on it: gravity (denoted by \(m\vecs g\)), the friction force \(\vecs f\), and the force exerted by the road \(\vecs N\).Let \(v_{max}\) denote the maximum speed the car can attain through the curve without skidding. In other words, \(v_{max}\) is the fastest speed at which the car can navigate the turn. When the car is traveling at this speed, the magnitude of the centripetal force is \[\| \vecs{F}_{cent} \| = \dfrac{m(v_{max})^2}{R}. \nonumber \] The next three questions deal with developing a formula that relates the speed \(v_{max}\) to the banking angle \(\theta\).
The coefficient of friction for a normal tire in dry conditions is approximately 0.7. Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.98. Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step.
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