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Chapter 12: Problem 36

Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. $$\mathbf{r}(t)=\langle 10 \cos t,-10 \sin t\rangle$$

Short Answer

Expert verified
Answer: The tangential component of acceleration is $\mathbf{a}_t = 0$, and the normal component of acceleration is $\mathbf{a}_n = 10$.

Step by step solution

01

Find the velocity vector

To find the velocity vector, we need to take the derivative of the given position vector function with respect to time, \(t\). The derivatives of the components of the position vector \(\mathbf{r}\) are: $$\frac{d}{dt} ( 10 \cos t) = -10 \sin t \quad \text{and} \quad \frac{d}{dt} (-10 \sin t) = -10 \cos t$$ So the velocity vector is given by: $$\mathbf{v}(t) = \langle -10 \sin t, -10 \cos t \rangle$$
02

Find the acceleration vector

To find the acceleration vector, we need to take the derivative of the velocity vector function with respect to time, \(t\). The derivatives of the components of the velocity vector \(\mathbf{v}\) are: $$\frac{d}{dt} (-10 \sin t) = -10 \cos t \quad \text{and} \quad \frac{d}{dt} (-10 \cos t) = 10 \sin t$$ So the acceleration vector is given by: $$\mathbf{a}(t) = \langle -10 \cos t, 10 \sin t \rangle$$
03

Compute the tangential component of acceleration

The tangential component of acceleration is given by the time rate of change of the magnitude of the velocity vector: $$\mathbf{a}_t = \frac{d||\mathbf{v}||}{dt}$$ Calculate the magnitude of the velocity vector: $$||\mathbf{v}|| = \sqrt{(-10 \sin t)^2 + (-10 \cos t)^2} = 10$$ Then take the derivative of the magnitude of velocity with respect to time: $$\frac{d||\mathbf{v}||}{dt} = \frac{d}{dt}(10) = 0$$ So the tangential component of acceleration is: $$\mathbf{a}_t = 0$$
04

Compute the normal component of acceleration

The normal component of acceleration is given by the following formula: $$\mathbf{a}_n = \frac{||\mathbf{v} \times \mathbf{a}||}{||\mathbf{v}||}$$ Calculate the cross product of the velocity and acceleration vectors: $$\mathbf{v} \times \mathbf{a} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -10 \sin t & -10 \cos t & 0 \\ -10 \cos t & 10 \sin t & 0 \end{vmatrix} = 0\mathbf{i} + 0\mathbf{j} + (-10 \sin t)^2 - (-10\cos t)^2 = 100\mathbf{k}$$ Now calculate the magnitude of the cross product: $$||\mathbf{v} \times \mathbf{a}|| = || 100\mathbf{k}|| = 100$$ Finally, compute the normal component of acceleration: $$\mathbf{a}_n = \frac{||\mathbf{v} \times \mathbf{a}||}{||\mathbf{v}||} = \frac{100}{10} = 10$$ In conclusion, the tangential and normal components of the acceleration vector are: $$\mathbf{a}_t = 0$$ $$\mathbf{a}_n = 10$$

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Most popular questions from this chapter

Trajectory with a sloped landing Assume an object is launched from the origin with an initial speed \(\left|\mathbf{v}_{0}\right|\) at an angle \(\alpha\) to the horizontal, where \(0 < \alpha < \frac{\pi}{2}\) a. Find the time of flight, range, and maximum height (relative to the launch point) of the trajectory if the ground slopes downward at a constant angle of \(\theta\) from the launch site, where \(0 < \theta < \frac{\pi}{2}\) b. Find the time of flight, range, and maximum height of the trajectory if the ground slopes upward at a constant angle of \(\theta\) from the launch site.

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle\)

The points \(P, Q, R,\) and \(S,\) joined by the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{x},\) are the vertices of a quadrilateral in \(\mathrm{R}^{3}\). The four points needn't lie in \(a\) plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that \(\mathbf{u}+\mathbf{v}=\mathbf{w}+\mathbf{x}\) b. Let \(m\) be the vector that joins the midpoints of \(P Q\) and \(Q R\) Show that \(\mathbf{m}=(\mathbf{u}+\mathbf{v}) / 2\) c. Let n be the vector that joins the midpoints of \(P S\) and \(S R\). Show that \(\mathbf{n}=(\mathbf{x}+\mathbf{w}) / 2\) d. Combine parts (a), (b), and (c) to conclude that \(\mathbf{m}=\mathbf{n}\) e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle-2,2\rangle, \mathbf{v}=\langle 2,1\rangle\)

Derive the formulas for time of flight, range, and maximum height in the case that an object is launched from the initial position \(\left\langle 0, y_{0}\right\rangle\) with initial velocity \(\left|\mathbf{v}_{0}\right|\langle\cos \alpha, \sin \alpha\rangle\).
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