Physics 221 - Spring 2002 - Exam 1
P221/S2002/Problem 1A
A particle starts at
% ~
0 m at
! ~ % shown for the time interval from
! ~
0 s to
! ~
8 s.
7
6
5
4
3
2
1
0
0 1 2 3 4 5 t ( in seconds)
6 7 8
# ²!³
, of its velocity is
Determine:
• (2 points) the position of the particle at
! ~
4 s:
%² ³ ~
15 m
(area under graph from
! ~
0 s to
! ~
5 s)
• (2 points) the velocity of the particle at
! ~
4 s:
# ² ³ ~
6 m/s
(read directly from graph)
• (2 points) the acceleration of the particle at
! ~
4 s:
4 s 0 (since
#
% is constant at
! ~
5 s).
• (2 points) the acceleration of the particle at
! ~
7 s:
² ³ ~ c
3 m/s)/s
~ c
• (3 points) the average velocity of the particle from
! ~
1 s to
! ~
4 s:
#
%,av
~
(15 m c
1 m)/3 s
~ %
• (3 points) the average acceleration of the particle from
! ~
2 s to
! ~
5 s:
(6 m/s c
4 m/s)/3 s
~
0.67 m/s .

P221/S2002/Problem 1B
A basketball player throws the basketball with an initial speed
#
0
at an angle above the horizontal, as shown. The ball travels to the hoop, located at a horizontal distance and at a height above the point of release. And it goes in! Swish!
(a) [4 points] Write down the general equations for the position components and
& was launched.
%²!³ ~ ²#
0
&²!³ ~ ²#
0
sin
³ ! c !
2
(b) [3 points] In terms of the given quantities (
#
0
, , , and ) and , if necessary, or
%²!³ ~ 3
. This occurs at time
! ~ 3°²#
0
(c) [2 points] What is the mathematical condition that the ball be at the correct height at the time you determined in part (b)?
At the time the ball reaches the basket, its height must be
& ~
.
(d) [5 points] Show that for the ball to go into the hoop, the initial speed must be given by
# 2
0
3
2 cos tan c°3³
Using the expression for the time derived in part (b), or
&²!³ ~ ~ ²#
0
sin
³²3°²#
0
cos
³ c ²3°²#
0
cos
³³ 2
~ 3
tan c
2
0
3 2
Rewriting this to solve for
#
0 we have c ~
3 2
2
# 2
0
cos
3 2
2 cos c³
.
² ³²
3 c°3³

(e) [3 bonus points] Show that as the ball goes through the hoop, the angle its velocity makes to the horizontal is given by tan 2 tan .
The angle is given by
#
#
&
%
# c !
#
%
0 c ²3°#
#
0
cos
0
cos
#
0 tan tan tan c
(2 tan c °3 ~ °3 c tan , using the expression from part (d).
Note that tan will actually be negative, since is below the horizontal and correspond to a negative angle..

P221/S2002/Problem 1C
A stone of mass attached to the end of a rope moves in a vertical circle of radius
9
, solely under the influence of gravity and the tensile force of the rope.
following points, expressing your answer in terms of known quantities and showing all your work. Begin each case by drawing a free-body force diagram for the stone.
(1) [2 points] at the lowest point: up, so
The tensile force is up, the gravitational force is down, and the centripetal acceleration is
; c ~ # °9
and
; ~ # °9 b
.
(2) [2 points] at the highest point:
The tensile force is down, the gravitational force is down, and the centripetal acceleration is also down, so
; b ~ # °9
and
(3) [5 points] when the rope is at a general angle to the vertical. What is the tension when
~
0° and
~
180°?
The tensile force of magnitude is directed toward the center of the circle, as is the centripetal acceleration
# °9
. The gravitational force also has a component of magnitude
cos in that direction. Thus
; b
cos
~ # °9
so
; ~ # °9 c
cos .
There is also a component sin of the gravitational force in the tangential direction; it produces a tangential acceleration which causes the speed of the particle to change.
At
~
0° we have
; ~ # °9 c
, the result for the highest point.
At
~
180° we have
; ~ # °9 b
, the result for the lowest point.
(b) [5 points] Suppose the maximum tension which the rope can sustain is
; max
. Where would it be most likely to break, and what is the maximum speed the stone could have at this without breaking the rope?
It would be most likely to break at the bottom, where the tension in the rope is the greatest.
The maximum speed would be given there by
; max
~ = °9 b
or
= ~ l 9; ° c 9
(c) [3 bonus points] What is the minimum speed the stone must have at the highest point to be able to move in a complete circle?
The minimum speed is that for which
; ~
0, or
# °9 c ~
0, or
# ~ # min
~ l 9À

Answer these questions (worth 2 points each) on the machine-gradeable form.
1. In scientific notation, retaining the correct number of significant figures:
4.24
d
10
4 c
2.4
d
10
3 ~
(A) 2.8
d
10
(D) 4.00
d
10
(B) 40 d
10
(E) 4.000
d
10
4
(C) 4.0
d
10 4
D is correct.
2. If |A b
B| = A 2 b
B then:
(A) A and B are perpendicular (B) The angle between A and B is 45°
(C) The angle between A and B is 60°
(D) A and B are parallel and in the same direction
(E) A and B are parallel and in opposite directions
(A) is correct.
3. A large truck collides head-on with a small car. During the collision
(A) the force the truck exerts on the car has a greater magnitude than the force the car exerts on the truck.
(B) the force the car exerts on the truck has a greater magnitude than the force the truck exerts on the car.
(C) the force the car exerts on the truck has the same magnitude as the force the truck exerts on the car.
(D) the truck exerts a force on the car but the car does not exert a force on the truck.
(E) neither vehicle exerts a force on the other
(C) is the only correct statement.
4. Let
W c
4 .0 j . Then the magnitude of 4 is
(A) 10 (B) 20 (C) 30 (D) 40 (E) 50
(B) is correct.

5. Which of the following is not true? The electric force
(A) is proportional to the inverse of the square of the distance between two charged particles.
(B) between an electron and a proton is much stronger than the gravitational force between them.
(C) between two protons separated by a distance is greater in magnitude than that between two electrons separated by a distance .
(D) can be either attractive or repulsive.
(E) is proportional to the product of the two electric charges involved.
(C) is the only false statement.
6. Two blocks of masses and
4
are pushed along a horizontal frictionless surface by a horizontal force
7W
, as shown in the diagram below. What is the magnitude of the force that either of these blocks exerts on the other?
(A)
7
(B) 1
7
2
7 TTS
1 2
--------------------------------------
(C) 2
7
1
(D)
1
2
7 b
2
(E)
1
1
7 b
2
(D) is correct
7. Which of the following five graphs is correct for a particle moving in a circle of radius at a constant angular speed of 10 rad/s?
# ~
so B is the correct diagram.

8. A dart is thrown horizontally toward X at 20 m/s as shown. It hits Y 0.10 seconds later.
The distance XY is
(A) 2 m (B) 1 m (C) 0.5 m (D) 0.1 m (E) 0.05 m
E is correct; it is the distance any object drops in 0.10 seconds:
(0.5)(10 m/s )(0.10 s) 2 ~
0.05 m.
9. Which of the following is NOT an example of accelerated motion?
(A) The vertical component of projectile motion on earth
(B) Circular motion at a constant speed
(C) A pendulum swinging back and forth
(D) The earth's motion about the sun
(E) The horizontal component of projectile motion on earth
E is the correct answer.
10. A boy pulls a wooden box of mass along a rough horizontal floor at constant speed by means of a force
7W
. The force diagram for the box is shown below. Which of the following must be true, where and
5
are, respectively, the magnitudes of the frictional and normal forces?
(A)
7 €
and
5 ~
(B)
7 ~
and
5 ~
(C)
7 €
and
5 
(D)
7 ~
and
5 €
W
§
¨
W
W
(E)
7 
and
5 ~
(B) is correct because the net force must equal zero, so the horizontal forces must have the same magnitudes and the vertical forces must also.