Dr. Fritz Wilhelm Homework Problems Pys.130
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Created on 8/14/2010 7:29:00 PM; C:\physics\130 lecture\ch 04 homwork.docx 8/14/2010
1. A motorist drives south at 20m/s for 3.00 min, then turns west and travels at 25m/s for 2
min, and finally travels northwest at 30 m/s for 1 min. For this 6 min trip find a) the
total vector displacement b) the average speed c)the average velocity. Let the positive xaxis point east. a) 4.87 km at 209º b) 23.3 m/s c) 13.5 m/s at 209º.
2. (5) A fish swimming in a horizontal plane has a velocity vi   4i  j 
m
at a point in the
s
ocean where its position relative to a rock is ri  10i  4 j  m . After the fish swims with
constant acceleration for 20s its velocity is v   20i  5 j 
m
s
a) What are the components of the acceleration? b) What is the direction of acceleration
with respect to unit vector i ? c) If the fish maintains constant acceleration, where is it at
t=25s, and in what direction is it moving?
3. (9) In a local bar, a customer slides an empty beer mug down the counter for a refill. It
slides off the counter and strikes the floor 1.40 m from the base of the counter which has
a height of 0.860 m.
a) with what velocity did the mug leave the counter?
b) What was the direction of the mug's velocity just before it hits the floor?
m
a) v  3.34 i b) -50.9º
s
4. (13) A projectile is fired in such a way that its horizontal range is equal to three times its
maximum height. What is the angle of projection?
53.1º
5. (15) A ball is tossed from an upper story window of a building. The ball is given an
initial velocity of 8.00 m/s at an angle 20º below the horizontal. It strikes the ground 3.00
s later.
a) How far horizontally from the base of the building does the ball strike the ground.
b) Find the height from which the ball was thrown.
c) How long does it take the ball to reach a point 10.0 m below the level of launching?
a) 22.6m b)52.3m c) 1.18s
Dr. Fritz Wilhelm Homework Problems Pys.130
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Created on 8/14/2010 7:29:00 PM; C:\physics\130 lecture\ch 04 homwork.docx 8/14/2010
6. (21) A soccer player kicks a rock horizontally off a 40.0 m high cliff into a pool of water.
If the player hears the sound of the splash 3.00 s later, what was the initial speed given to
the rock? Assume the speed of sound to be 343 m/s.
9.91 m/s
7. (27) Young David who slew Goliath experimented with a sling. He found that he could
revolve a sling of length 0.600 m at a rate of 8.00 revolutions per second. If he increased
the length to 0.900m he achieved a rate of 6.00 rps.
a) Which rate gives the greater speed to the rock at the end of the sling?
b) What is the centripetal acceleration at the rate of 8.00 ps?
c) What is the centripetal acceleration at 6.00 rps.
8. Make a drawing to scale of a vector A with length 5in and making the angle of 30˚ with
the horizontal positive x-axis.
a) Draw the components of this vector and verify that the x-component is equal to Acosθ
and the y component is equal to Asinθ.
b) At the end of this vector A draw a second vector perpendicular to the first one,
pointing to the left as seen from the first vector. Show that this vector has the x
component -Asinθ and the y-component +Acosθ.
9. Prove that the time derivative of the unit vector in the radial direction is proportional to
the unit vector in the tangential direction. The definition of the radial unit vector is given
in the next problem, and the tangential unit vector is given in
Cartesian coordinates by u  sint ,cos t . (Memorize these definitions!)
10. Prove that both unit vectors have indeed magnitude 1 by actually calculating their
magnitude, using the definition for the magnitude of a vector by:
A  A  Ax2  Ay2
Dr. Fritz Wilhelm Homework Problems Pys.130
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Created on 8/14/2010 7:29:00 PM; C:\physics\130 lecture\ch 04 homwork.docx 8/14/2010
11. Starting with the vector
r (t )  rur where r is the magnitude and ur is the unit vector <cost,sint >
take its derivative twice to find its acceleration in polar coordinates directly. Prove that
if ω is constant, you obtain the acceleration ac   2 rur
Hint: You must use the chain rule and the product rule for derivatives.
12. Prove that the two unit vectors in the radial and in the tangential direction of a particle
moving in a circle are perpendicular to each other.
ur  cost,sint ; u  sint,cost Recall that the scalar product of two vectors is
0 if they are perpendicular to each other.
13. Draw a circle and find the direction of the vector  which is implicitly defined by the
cross product v=  r , with v being the tangential velocity, and r being the radial vector
pointing from the center of the circle to its circumference.
Answer: the vector is perpendicular to the circle.
14. (29) A train slows down as it enters a sharp horizontal turn, slowing from 90.0 to 50.0
km/h in 15 s. The radius of the bent is 150m. Compute the acceleration in the moment the
train reaches the speed of 50.0km/h. Assume that it maintains its deceleration.
1.48m/s2. inward at the angle of 29.9˚ backward.
15. (31) A particle moves clockwise in a circle of radius 2.50 m. At a certain instant its
resultant acceleration is 15.0m/s2 and makes an angle of 30˚ with the radius.
a) Find the radial acceleration, (13.0 m/s2); b) the speed of the particle (5.70 m/s) and c)
the tangential acceleration at this instant (7.50 m/s2).
16. (32) A racecar starts from rest on a circular track. It increases its speed at a constant rate
as it goes around the track. Find the angle the total acceleration makes with the radius
after the car completes one full circle. 1/4π
17. (35) A river has a steady speed of 0.500m/s. A student swims upstream for 1km and
swims back to the starting point. If the student can swim at 1.20m/s in still water, how
long does the trip take. Compare this result to the situation where the student would swim
back and forth in still water.
2.02E3s, 21.0% longer.
Dr. Fritz Wilhelm Homework Problems Pys.130
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Created on 8/14/2010 7:29:00 PM; C:\physics\130 lecture\ch 04 homwork.docx 8/14/2010
18. (39) A girl is riding on the flat car of a train traveling along a straight horizontal track at
constant speed of 10.0 m/s. The student throws a ball into the air along a path that he
judges to make an initial angle of 60.0 degrees with the horizontal and to be in line with
the track. A boy who is standing on the ground nearby observes the ball to rise vertically.
How high does he see the ball rise? Answer: 15.3m
19. Particle A moves to the left with a velocity of 0.500c; another particle B moves to the right
with a velocity of 0.900c. Find the relative velocity of particle B with respect to particle A.
Use relativistic definitions: 0.966c