1.5E: Exercises

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.

Answer

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

Answer

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

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:

Answer

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

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) $\vecs r(t)=⟨e^t,e^{−t}⟩$

Answer

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

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?

Answer

$t = 4$

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.

Answer

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

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

Answer

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

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

Answer

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

b. The speed of the particle

c. The acceleration of the particle

Answer

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

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.

Answer

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

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?

Answer

$44.185$ sec

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

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

Answer

$t=88.37$ sec

d. What is the maximum range?

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

Answer

$t=88.37$ 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.

Answer

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.

Answer

$v=42.16$ 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 $t=1$.

Answer

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

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:

Answer

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

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

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

Answer

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

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

29) $\vecs r(t)=⟨6t,\,3t^2,\,2t^3⟩$

Answer

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

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

Answer

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

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 $t=0$.

Answer

$a_\vecs{T}=0, \quad a_\vecs{N}=aω^2$

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 $t=1.5$.

Answer

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, $\vecs F=m\vecs a$.)

Answer

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

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.

Answer

$10.94$ 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.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.