12.3E: Exercises for Section 12.3
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Motion in Space
1) Given ⇀r(t)=(3t2−2)ˆi+(2t−sint)ˆj,
a. find the velocity of a particle moving along this curve.
b. find the acceleration of a particle moving along this curve.

- Answer:
- a. ⇀v(t)=6tˆi+(2−cost)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)=e−tˆi+t2ˆj+tantˆk
3) ⇀r(t)=⟨3cost,3sint,t2⟩
- Answer:
- ⇀v(t)=−3sintˆi+3costˆj+2tˆk
⇀a(t)=−3costˆi−3sintˆj+2ˆk
Speed(t)=‖⇀v(t)‖=√9+4t2
4) ⇀r(t)=t5ˆi+(3t2+2t−5)ˆj+(3t−1)ˆk
5) ⇀r(t)=2costˆj+3sintˆk. The graph is shown here:
- Answer:
- ⇀v(t)=−2sintˆj+3costˆk
⇀a(t)=−2costˆj−3sintˆk
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)=⟨t2−1,t⟩
7) ⇀r(t)=⟨et,e−t⟩
- Answer:
- ⇀v(t)=⟨et,−e−t⟩,
⇀a(t)=⟨et,e−t⟩,
‖⇀v(t)‖=√e2t+e−2t
8) ⇀r(t)=⟨sint,t,cost⟩. The graph is shown here:
9) The position function of an object is given by ⇀r(t)=⟨t2,5t,t2−16t⟩. At what time is the speed a minimum?
- Answer:
- t=4
10) Let ⇀r(t)=rcosh(ωt)ˆi+rsinh(ωt)ˆj. Find the velocity and acceleration vectors and show that the acceleration is proportional to ⇀r(t).
11) Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid ⇀r(t)=(ωt−sin(ωt))ˆi+(1−cos(ωt))ˆj, where ω 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:
- ⇀v(t)=(ω−ωcos(ωt))ˆi+(ωsin(ωt))ˆj
⇀a(t)=(ω2sin(ωt))ˆi+(ω2cos(ωt))ˆj
speed(t)=√(ω−ωcos(ωt))2+(ωsin(ωt))2=√ω2−2ω2cos(ωt)+ω2cos2(ωt)+ω2sin2(ωt)=√2ω2(1−cos(ωt))
12) A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector ⇀r(t)=(3cost)ˆi+(3sint)ˆj+t2ˆk. 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:
- ‖⇀v(t)‖=√9+4t2
c. The times, if any, at which the glider’s acceleration is orthogonal to its velocity
13) Given that ⇀r(t)=⟨e−5tsint,e−5tcost,4e−5t⟩ is the position vector of a moving particle, find the following quantities:
a. The velocity of the particle
- Answer:
- ⇀v(t)=⟨e−5t(cost−5sint),−e−5t(sint+5cost),−20e−5t⟩
b. The speed of the particle
c. The acceleration of the particle
- Answer:
- ⇀a(t)=⟨e−5t(−sint−5cost)−5e−5t(cost−5sint),−e−5t(cost−5sint)+5e−5t(sint+5cost),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 ⇀r(t), given that ⇀a(t)=ˆi+etˆj,⇀v(0)=2ˆj, and ⇀r(0)=2ˆi.
16) Find ⇀r(t) given that ⇀a(t)=−32ˆj,⇀v(0)=600√3ˆi+600ˆj, and ⇀r(0)=⇀0.
17) The acceleration of an object is given by ⇀a(t)=tˆj+tˆk. The velocity at t=1 sec is ⇀v(1)=5ˆj and the position of the object at t=1 sec is ⇀r(1)=0ˆi+0ˆj+0ˆk. Find the object’s position at any time.
- Answer:
- ⇀r(t)=0ˆi+(16t3+4.5t−143)ˆj+(16t3−12t+13)ˆk
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.