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Exercises for Section 13.3

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Motion in Space

1) Given r(t)=(3t22)ˆi+(2tsint)ˆj,

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

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

CNX_Calc_Figure_13_04_205.jpg
Answer:
a. v(t)=6tˆi+(2cost)ˆi
b. a(t)=6ˆi+sintˆi

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

2) r(t)=etˆi+t2ˆj+tantˆk

3) r(t)=3cost,3sint,t2

Answer:
v(t)=3sintˆi+3costˆj+2tˆk
a(t)=3costˆi3sintˆj+2ˆk
Speed(t)=v(t)=9+4t2

4) r(t)=t5ˆi+(3t2+2t5)ˆj+(3t1)ˆk

5) r(t)=2costˆj+3sintˆk. The graph is shown here:

Figure13_4_Ex5.jpeg

Answer:
v(t)=2sintˆj+3costˆk
a(t)=2costˆj3sintˆk
Speed(t)=v(t)=4sin2t+9cos2t=4+5cos2t

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

6) r(t)=t21,t

7) r(t)=et,et

Answer:
v(t)=et,et,
a(t)=et,et,
v(t)=e2t+e2t

8) r(t)=sint,t,cost. The graph is shown here:

CNX_Calc_Figure_13_04_207.jpg

9) The position function of an object is given by r(t)=t2,5t,t216t. 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:

CNX_Calc_Figure_13_04_201.jpg

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:

CNX_Calc_Figure_13_04_208.jpg

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.

 

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.


Exercises for Section 13.3 is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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