
# 13.4E: Exercises for Section 13.4


1) Given $$\vecs r(t)=(3t^2−2)\,\hat{\mathbf{i}}+(2t−\sin t)\,\hat{\mathbf{j}}$$,

a. find the velocity of a particle moving along this curve.

b. find the acceleration of a particle moving along this curve.

a. $$\vecs v(t)=6t\,\hat{\mathbf{i}}+(2−\cos t)\,\hat{\mathbf{i}}$$
b. $$\vecs a(t)=6\,\hat{\mathbf{i}}+\sin t\,\hat{\mathbf{i}}$$

In questions 2 - 5, given the position function, find the velocity, acceleration, and speed in terms of the parameter $$t$$.

2) $$\vecs r(t)=e^{−t}\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}+\tan t\,\hat{\mathbf{k}}$$

3) $$\vecs r(t)=⟨3\cos t,\,3\sin t,\,t^2⟩$$

$$\vecs v(t)=-3\sin t\,\hat{\mathbf{i}}+3\cos t\,\hat{\mathbf{j}}+2t\,\hat{\mathbf{k}}$$
$$\vecs a(t)=-3\cos t\,\hat{\mathbf{i}}-3\sin t\,\hat{\mathbf{j}}+2\,\hat{\mathbf{k}}$$
$$\text{Speed}(t) = \|\vecs v(t)\| = \sqrt{9 + 4t^2}$$

4) $$\vecs r(t)=t^5\,\hat{\mathbf{i}}+(3t^2+2t- 5)\,\hat{\mathbf{j}}+(3t-1)\,\hat{\mathbf{k}}$$

5) $$\vecs r(t)=2\cos t\,\hat{\mathbf{j}}+3\sin t\,\hat{\mathbf{k}}$$. The graph is shown here:

$$\vecs v(t)=-2\sin t\,\hat{\mathbf{j}}+3\cos t\,\hat{\mathbf{k}}$$
$$\vecs a(t)=-2\cos t\,\hat{\mathbf{j}}-3\sin t\,\hat{\mathbf{k}}$$
$$\text{Speed}(t) = \|\vecs v(t)\| = \sqrt{4\sin^2 t+9\cos^2 t}=\sqrt{4+5\cos^2 t}$$

In questions 6 - 8, find the velocity, acceleration, and speed of a particle with the given position function.

6) $$\vecs r(t)=⟨t^2−1,t⟩$$

7) $$\vecs r(t)=⟨e^t,e^{−t}⟩$$

$$\vecs v(t)=⟨e^t,−e^{−t}⟩$$,
$$\vecs a(t)=⟨e^t, e^{−t}⟩,$$
$$\|\vecs v(t)\| = \sqrt{e^{2t}+e^{−2t}}$$

8) $$\vecs r(t)=⟨\sin t,t,\cos t⟩$$. The graph is shown here:

9) The position function of an object is given by $$\vecs r(t)=⟨t^2,5t,t^2−16t⟩$$. At what time is the speed a minimum?

$$t = 4$$

10) Let $$\vecs r(t)=r\cosh(ωt)\,\hat{\mathbf{i}}+r\sinh(ωt)\,\hat{\mathbf{j}}$$. Find the velocity and acceleration vectors and show that the acceleration is proportional to $$\vecs r(t)$$.

11) Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid $$\vecs r(t)=(ωt−\sin(ωt))\,\hat{\mathbf{i}}+(1−\cos(ωt))\,\hat{\mathbf{j}}$$, where $$\omega$$ is the angular velocity of the circle and $$b$$ is the radius of the circle:

Find the equations for the velocity, acceleration, and speed of the particle at any time.

$$\vecs v(t)=(ω−ω\cos(ωt))\,\hat{\mathbf{i}}+(ω\sin(ωt))\,\hat{\mathbf{j}}$$
$$\vecs a(t)=(ω^2\sin(ωt))\,\hat{\mathbf{i}}+(ω^2\cos(ωt))\,\hat{\mathbf{j}}$$
\begin{align*} \text{speed}(t) &= \sqrt{(ω−ω\cos(ωt))^2 + (ω\sin(ωt))^2} \\ &= \sqrt{ω^2 - 2ω^2 \cos(ωt) + ω^2\cos^2(ωt) + ω^2\sin^2(ωt)} \\ &= \sqrt{2ω^2(1 - \cos(ωt))} \end{align*}

12) A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector $$\vecs r(t)=(3\cos t)\,\hat{\mathbf{i}}+(3\sin t)\,\hat{\mathbf{j}}+t^2\,\hat{\mathbf{k}}$$. The path is similar to that of a helix, although it is not a helix. The graph is shown here:

Find the following quantities:

a. The velocity and acceleration vectors

b. The glider’s speed at any time

$$\|\vecs v(t)\|=\sqrt{9+4t^2}$$

c. The times, if any, at which the glider’s acceleration is orthogonal to its velocity

13) Given that $$\vecs r(t)=⟨e^{−5t}\sin t,\, e^{−5t}\cos t,\, 4e^{−5t}⟩$$ is the position vector of a moving particle, find the following quantities:

a. The velocity of the particle

$$\vecs v(t)=⟨e^{−5t}(\cos t−5\sin t),\, −e^{−5t}(\sin t+5\cos t),\, −20e^{−5t}⟩$$

b. The speed of the particle

c. The acceleration of the particle

$$\vecs a(t)=⟨e^{−5t}(−\sin t−5\cos t)−5e^{−5t}(\cos t−5\sin t), \; −e^{−5t}(\cos t−5\sin t)+5e^{−5t}(\sin t+5\cos t),\; 100e^{−5t}⟩$$

14) Find the maximum speed of a point on the circumference of an automobile tire of radius $$1$$ ft when the automobile is traveling at $$55$$ mph.

15) Find the position vector-valued function $$\vecs r(t)$$, given that $$\vecs a(t)=\hat{\mathbf{i}}+e^t \,\hat{\mathbf{j}}, \quad \vecs v(0)=2\,\hat{\mathbf{j}}$$,   and   $$\vecs r(0)=2\,\hat{\mathbf{i}}$$.

16) Find $$\vecs r(t)$$ given that $$\vecs a(t)=−32\,\hat{\mathbf{j}}, \vecs v(0)=600\sqrt{3} \,\hat{\mathbf{i}}+600\,\hat{\mathbf{j}}$$, and $$\vecs r(0)=\vecs 0$$.

17) The acceleration of an object is given by $$\vecs a(t)=t\,\hat{\mathbf{j}}+t\,\hat{\mathbf{k}}$$. The velocity at $$t=1$$ sec is $$\vecs v(1)=5\,\hat{\mathbf{j}}$$ and the position of the object at $$t=1$$ sec is $$\vecs r(1)=0\,\hat{\mathbf{i}}+0\,\hat{\mathbf{j}}+0\,\hat{\mathbf{k}}$$. Find the object’s position at any time.

$$\vecs r(t)=0\,\hat{\mathbf{i}}+\left(\frac{1}{6}t^3+4.5t−\frac{14}{3}\right)\,\hat{\mathbf{j}}+\left(\frac{1}{6}t^3−\frac{1}{2}t+\frac{1}{3}\right)\,\hat{\mathbf{k}}$$

## Projectile Motion

18) A projectile is shot in the air from ground level with an initial velocity of $$500$$ m/sec at an angle of 60° with the horizontal.

a. At what time does the projectile reach maximum height?

$$44.185$$ sec

b. What is the approximate maximum height of the projectile?

c. At what time is the maximum range of the projectile attained?

$$t=88.37$$ sec

d. What is the maximum range?

e. What is the total flight time of the projectile?

$$t=88.37$$ sec

19) A projectile is fired at a height of $$1.5$$ m above the ground with an initial velocity of $$100$$ m/sec and at an angle of 30° above the horizontal. Use this information to answer the following questions:

a. Determine the maximum height of the projectile.

b. Determine the range of the projectile.

The range is approximately $$886.29$$ m.

20) A golf ball is hit in a horizontal direction off the top edge of a building that is 100 ft tall. How fast must the ball be launched to land $$450$$ ft away?

21) A projectile is fired from ground level at an angle of 8° with the horizontal. The projectile is to have a range of $$50$$ m. Find the minimum velocity (speed) necessary to achieve this range.

$$v=42.16$$ m/sec

22) Prove that an object moving in a straight line at a constant speed has an acceleration of zero.

## Finding Components of Acceleration & Kepler's Laws

23) Find the tangential and normal components of acceleration for $$\vecs r(t)=t^2\,\hat{\mathbf{i}}+2t \,\hat{\mathbf{j}}$$ when $$t=1$$.

$$a_\vecs{T}=\sqrt{2}, \quad a_\vecs{N}=\sqrt{2}$$

In questions 24 - 30, find the tangential and normal components of acceleration.

24) $$\vecs r(t)=⟨\cos(2t),\,\sin(2t),1⟩$$

25) $$\vecs r(t)=⟨e^t \cos t,\,e^t\sin t,\,e^t⟩$$. The graph is shown here:

$$a_\vecs{T}=\sqrt{3}e^t, \quad a_\vecs{N}=\sqrt{2}e^t$$

26) $$\vecs r(t)=⟨\frac{2}{3}(1+t)^{3/2}, \,\frac{2}{3}(1-t)^{3/2},\,\sqrt{2}t⟩$$

27) $$\vecs r(t)=\left\langle 2t,\,t^2,\,\dfrac{t^3}{3}\right\rangle$$

$$a_\vecs{T}=2t, \quad a_\vecs{N}=2$$

28) $$\vecs r(t)=t^2\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}+t^3\,\hat{\mathbf{k}}$$

29) $$\vecs r(t)=⟨6t,\,3t^2,\,2t^3⟩$$

$$a_\vecs{T}=\dfrac{6t +12t^3}{\sqrt{1+t^2+t^4}}, \quad a_\vecs{N}=6\sqrt{\dfrac{1+4t^2+t^4}{1+t^2+t^4}}$$

30) $$\vecs r(t)=3\cos(2πt)\,\hat{\mathbf{i}}+3\sin(2πt)\,\hat{\mathbf{j}}$$

$$a_\vecs{T}=0, \quad a_\vecs{N}=12\pi^2$$

31) Find the tangential and normal components of acceleration for $$\vecs r(t)=a\cos(ωt)\,\hat{\mathbf{i}}+b\sin(ωt)\,\hat{\mathbf{j}}$$ at $$t=0$$.

$$a_\vecs{T}=0, \quad a_\vecs{N}=aω^2$$

32) Suppose that the position function for an object in three dimensions is given by the equation $$\vecs r(t)=t\cos(t)\,\hat{\mathbf{i}}+t\sin(t)\,\hat{\mathbf{j}}+3t\,\hat{\mathbf{k}}$$.

a. Show that the particle moves on a circular cone.

b. Find the angle between the velocity and acceleration vectors when $$t=1.5$$.

c. Find the tangential and normal components of acceleration when $$t=1.5$$.

c. $$a_\vecs{T}=0.43\,\text{m/sec}^2, \quad a_\vecs{N}=2.46\,\text{m/sec}^2$$

33) The force on a particle is given by $$\vecs f(t)=(\cos t)\,\hat{\mathbf{i}}+(\sin t)\,\hat{\mathbf{j}}$$. The particle is located at point $$(c,0)$$ at $$t=0$$. The initial velocity of the particle is given by $$\vecs v(0)=v_0\,\hat{\mathbf{j}}$$. Find the path of the particle of mass $$m$$. (Recall, $$\vecs F=m\vecs a$$.)

$$\vecs r(t)=\left(\dfrac{-\cos t}{m}+c+\frac{1}{m}\right)\,\hat{\mathbf{i}}+\left(\dfrac{−\sin t}{m}+\left(v_0+\frac{1}{m}\right)t\right)\,\hat{\mathbf{j}}$$

34) An automobile that weighs $$2700$$ lb makes a turn on a flat road while traveling at $$56$$ ft/sec. If the radius of the turn is $$70$$ ft, what is the required frictional force to keep the car from skidding?

35) Using Kepler’s laws, it can be shown that $$v_0=\sqrt{\dfrac{2GM}{r_0}}$$ is the minimum speed needed when $$\theta=0$$ so that an object will escape from the pull of a central force resulting from mass $$M$$. Use this result to find the minimum speed when $$\theta=0$$ for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of $$300$$ km above Earth’s surface.

$$10.94$$ km/sec
36) Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that $$a=39.5$$ A.U.