1. A particle travels along a path in the x-y plane. Its position at any instant is given by
a) Calculate the position r, the velocity v, and acceleration a of the particle at t=2 s.
b) Consider the motion of the particle from t= 0 s to t= 2 s. What are the displacement and average velocity of the particle over this time interval?
2. A missile is shot horizontally from the top of a 500 m cliff with an initial speed of 300 m/s.
a) Find the time it takes for the missile to hit the ground.
b) What is the range of the missile?
c) Calculate the velocity of the missile just before it hits the ground.
d) If the missile hits the ground and bounces up at an angle of 30 degrees with a speed of 200 m/s, how far away from the point of impact will it land?
3. A good baseball pitcher can throw a fastball at about 132 ft/s (90 mph).
Assuming a constant acceleration for the ball, if he starts the ball's
motion at rest 1.5 ft behind his head and releases the ball at its final
speed 2.5 ft in front of his head (neglecting any motion of his body),
what is the acceleration of the ball during the throw?
4. A baseball is thrown from the ground into the air with an initial velocity
v = 20î+10j m/s.
a) To what maximum height will the ball rise?
b) What will be the total horizontal distance travelled by the ball?
c) What is the velocity of the ball the instant before it hits the ground?
5. You get to your apartment from school after a long day and try to turn on your radio which is broken. In frustration you kick the radio out of the window which is 20 m from the ground so that the radio is moving at 30î m/s.
a) How long does it take the radio to hit the ground?
b) How far (horizontally) from window does the ball land?
c) What is the velocity of the radio just before it hits the ground?
d) How long does it take for the radio to fall halfway to the ground?
6. In the problem above, remember you kicked a radio so that it went flying out the window. Assume the radio is sitting in the window initially and it moves 0.25 m horizontally until it is traveling the 30î m/s. Answer the following questions if the radio moves under constant acceleration.
a) What is the acceleration of the radio?
b) How long did it take for the radio to reach its final velocity?
c) What is the speed of the radio after it has been accelerated for 0.1 m after it is first in contact with your foot?
7. A bullet is shot into a block of wood and decelerates uniformly to a stop after penetrating a distance of 0.18 meters into the wood. Just before entering the obstacle, the bullet is traveling 450 m/s.
a) What is the acceleration of the bullet?
b) How much time does it take for the bullet to come to rest?
c) What is the average velocity of the bullet?
d) How fast is the bullet going after it has traveled a distance of 0.09 m into the wood block?
8. Consider the act of shooting an arrow with a bow. As you set the arrow in place, you pull straight back on the bow a distance of 0.35 m. Releasing it from rest, the arrow is uniformly accelerated by the bow until it reaches a speed of 30 m/s as it leaves.
a) Calculate the acceleration of the arrow in the bow.
b) After you release it, how much time does it take for the arrow to reach the final speed?
c) How far has the arrow traveled when the speed is 20 m/s?
d) How fast is the arrow going when the bow pushes it for 0.05 m?9. In this problem you are asked to think about the action of landing an airplane on a runway. The airplane hits the ground travelling 200 m/s on a runway which is 1000 m long.
a) Find the minimum acceleration necessary to stop the plane before it runs out of room.
b) How much time would it take to stop the plane with this acceleration?
c) Under the conditions found in part a), how fast is the plane going halfway down the runway?
d) If you are taking off from the same runway, find the acceleration needed to reach a final take-off speed of 500 m/s?
10. A particle is constrained to travel in a plane so that
its position is described by r = (17
m/s2)t2î
+ [(20m/s)t - (6 m/s4)t4
]
. Keep all the digits as significant figures in your answers; in other words, do
not round off at all!
a) What are the position and velocity after 5 seconds?
b) Consider the motion of the particle during the time interval from t=0 s to t=4 s? What are the displacement and average velocity over this time interval.
c) Calculate the average velocity and average acceleration over the interval from 0 seconds to 4 s?
d) Find the position, velocity and acceleration at t=8 s.
11. A particle travels in the x-y plane with position
described by r = (90m/s3)t3î
+ [180m - (8 m/s5)t5
]
.
a) Consider the motion of the particle during the time interval from t=0 s to t=3s? What are the displacement, average velocity and average acceleration over this interval.
b) What are the position and instantaneous velocity after 10 s?
c) Calculate the displacement and average velocity over the interval from 0 seconds to 10 s?
d) Find the acceleration at t=10 s, and also calculate the average acceleration for the time interval t=0 s to t=10s.
12. Consider the act of shooting an arrow with a bow. As you set the arrow in place, you pull straight back on the bow a distance of 0.35 m. Releasing it from rest, the arrow is uniformly accelerated by the bow until it reaches a speed of 30 m/s as it leaves.
a) Calculate the acceleration of the arrow in the bow.
b) After you release it, how much time does it take for the arrow to reach the final speed?
c) How far has the arrow traveled when the speed is 20 m/s?
d) How fast is the arrow going when the bow pushes it for 0.05 m?
13. A particle travels in the x-y plane with position
described by r = (90m/s3)t3î
+ [180m - (8 m/s5)t5
]
.
a) Consider the motion of the particle during the time interval from t=0 s to t=3s? What are the displacement, average velocity and average acceleration over this interval.
b) What are the position and instantaneous velocity after 10 s?
c) Calculate the displacement and average velocity over the interval from 0 seconds to 10 s?
d) Find the acceleration at t=10 s, and also calculate the average acceleration for the time interval t=0 s to t=10s.
14. Consider the act of shooting an arrow with a bow. As you set the arrow in place, you pull straight back on the bow a distance of 0.35 m. Releasing it from rest, the arrow is uniformly accelerated by the bow until it reaches a speed of 30 m/s as it leaves.
a) Calculate the acceleration of the arrow in the bow.
b) After you release it, how much time does it take for the arrow to reach the final speed?
c) How far has the arrow traveled when the speed is 20 m/s?
d) How fast is the arrow going when the bow pushes it for 0.05 m?
15. A golfer hits a golf ball with an initial speed of 30 m/s at an angle of 35o with the ground. The ball flies over the level ground and lands at the same height from which it was hit.
a) How long does it take for the ball to reach its highest point?
b) What is the range of the ball?
c) Find the velocity of the ball just before it hits the ground.
d) After the ball hits the ground it rolls a short distance and then rolls off the horizontal surface of a cliff which is 23 meters above the ground below. If the ball slows down to 12 m/s just before it goes off the cliff, how far from the base of the cliff does the ball land?
16. A dart is fired from a toy dart gun with an initial velocity vo=14 m/sî. Answer the following questions assuming the dart is fired from the top of a 9 meter tower.
a) How long does it take for the dart to reach the ground?
b) What is the range of the dart?
c) Using standard Cartesian coordinates with the origin at the top of the tower, find the displacement of the dart for its trip through the air.
d) What is the velocity of the dart after it has fallen a distance of 4 m from the top of the tower?
e) If a stronger spring is put in the gun so the initial speed of the dart is 19 m/s, then how long does it take for the dart to reach the ground?
17. You get home after completing your physics test one Friday afternoon. Since you think you just aced the exam, you decide to open a champagne bottle in celebration. Tilting the bottle at an angle of 35o, the cork comes flying out with an initial velocity of 8 m/s.
a) Find the horizontal and vertical components of the initial velocity.
b) How much time does it take for the cork to reach the highest point in its trajectory?
c) How high did it go?
d) What is the range of the cork, assuming it lands at the same height you are holding the bottle?
e) Find the velocity and position of the cork 0.6 seconds after the cork leaves the bottle, assuming an origin at the top of the bottle.
18. A pitcher throws a fastball toward home plate. He throws the ball horizontally with a speed of 30 m/s a distance of 1.5 m above the ground. Answer the following questions assuming the ball misses the catcher and falls to the ground unimpeded.
a) How long does it take for the ball to hit the ground?
b) What is the range of the ball?
c) Calculate the velocity of the ball just before it hits the ground.
d) Find the average velocity of the ball as it travels from the pitcher to the ground.19. One day you notice a child swinging on a swing in the playground. As she swings to the highest point a distance of 2.8 m above the ground, she manages to jump off the swing horizontally going 3.5 m/s (it takes a little bit of skill on her part).
a) What is the velocity of the child just before she hits the ground?
b) How much times does it take from when she jumps to when she hits the ground?
c) Calculate the range of the child.
d) What is the velocity of the child when she has fallen halfway to the ground?
e) How much time does it take for her to reach the halfway point?
f) How much time does it take for her to hit the ground if she leaves the swing at the highest point from rest?