Momentum
From Old AP’s
1. (1974) An unstable nucleus has mass M and is initially at rest. It ejects a particle of
mass m with speed Vo. The remaining nucleus recoils in the opposite direction with a
speed:
a. Vo
c.
m
Vo
(m  M )
e.
m
V0
( M  m)
b.
m
Vo
M
d.
(m  M )
V0
m
*E. Since the nucleus begins at rest, the total momentum of the system is zero. Draw
before and after diagrams:
2. (1984) A system consists of two objects having masses M1 and M2 (M1 < M2). The
objects are connected by a massless string, hung over a pulley as shown above, and then
released. When the speed of each object is V, the magnitude of the total linear
momentum of the system is:
a. (M1 + M2)V
b. (M2 - M1) V
c.
(M 1  M 2 )
V
2
d.
(M 2  M 1 )
V
2
e. M2V
*B. Remember momentum is a vector, so M1 has positive momentum and M2 has
negative momentum (defining up as positive).
p y  M 1V  M 2V  ( M 1  M 2 )V
This is where the problem gets a little tricky.
Since M 2  M 1 , this value is negative so the MAGNITUDE of the total momentum
is the negative of this equation.
Note : it makes sense that the system is gaining momentum in the negative - Y direction
becuase the external force of gravity is directed down.
3. A ball of mass M and speed V collides head-on with a ball of mass 2M and speed V/2,
as shown above. If the two balls stick together, their speed after the collision is:
a. 0.
b. V/2
c.
2
V
2
d.
3
V
2
e.
2
V
3
* A. Remember: velocity going left is negative!!!
Pi  Pf
MV  (2M)(Vf  0
V
)  (3M)(V f )
2
4. A toy cannon is fixed to a small cart and both move to the right with speed V along a
straight track as shown above. The cannon points in the direction of motion. When the
cannon fires a projectile the cart and cannon are brought to rest. If M is the mass of
the cart and cannon combined without the projectile, and m is the mass of the projectile,
what is the speed of the projectile relative to the ground immediately after it is fired?
a.
M
V
m
b.
( M  m)
V
m
c.
( M  m)
V
m
d.
m
V
M
e.
m
V
( M  m)
* B. Conserve momentum
Pi  Pf
( M  m)V  M(0)  mV f
Vf 
( M  m)
V
m
Questions 5 and 6:
A 4 kg mass has a speed of 6 m/s on a horizontal frictionless surface, as shown above.
The mass collides head-on and elastically with an identical 4 kg mass initially at rest.
The second 4 kg mass then collides head-on and sticks to a third 4 kg mass initially at
rest.
5. The final speed of the first 4 kg mass is:
a. 0 m/s.
b. 2 m/s.
c. 3 m/s.
d. 4 m/s.
e. 6 m/s.
* A. Remember the batman demo. When two objects with equal mass collide elastically,
they trade velocities.
6. The final speed of the two 4 kg masses that stick together is:
a. 0 m/s.
b. 2 m/s.
c. 3 m/s.
d. 4 m/s.
e. 6 m/s.
* C. Since momentum is conserved, if the mass doubles, the speed is cut in half.
7. Two pucks are attached by a stretched spring and are initially at rest on a frictionless
surface, as shown above. The pucks are then released simultaneously. If puck I has three
times the mass of puck II, which of the following quantities is the same for both pucks as
the spring pulls the pucks toward each other?
a. Speed.
b. Velocity.
c. Acceleration.
d. Kinetic energy.
e. Magnitude of momentum
* E. Since the total momentum of the system was zero to begin with and since the spring
is an internal force, momentum must be conserved which means the two pucks must have
equal and opposite momenta.
8. (1974) A cart of mass 2M has a velocity Vo before it strikes another cart of mass 3M
at rest. The two carts stick together and move off together with a velocity of:
a.
V0
3
b.
2
V0
5
c.
2
V0
5
d.
3
V0
5
e.
2
V0
3
* B. Conserve momentum:
Pi  Pf
(2m)(Vo )  0  (5m)V f
2
V f  V0
5
9. (1993) An object of mass m is moving with speed Vo to the right on a horizontal
frictionless surface, as shown below, when it explodes into two pieces. Subsequently,
one piece of mass 2/5 m moves with a speed of Vo/2 to the left. The speed of the other
piece of the object is:
a.
V0
2
b.
4V0
3
c.
7V0
5
d.
3V0
2
e. 2Vo
* E. Remember that going left means negative velocity!!!
Pi  Pf
V
2
3
mVo  ( m)(  0 )  ( m)V f
5
2
5
V0  -
(If one piece is 2/5m, the other is 3/5 m)
V0
3
 Vf
5
5
5V0  - V0  3Vf
So
Vf  2V0
From Princeton Review Book
10. Two objects, one of mass 3 kg and moving with a speed of 2 m/s and the other of
mass 5 kg and a speed of 2 m/s, move toward each other and collide head-on. If the
collision is perfectly inelastic, find the speed of the objects after the collision.
a. 0.25 m/s.
b. .5 m/s.
c. 0.75 m/s
d. 1 m/s
e. 2 m/s.
* B. Conserve momentum—be careful!! Since they didn’t draw a diagram for you, it
might be tempting to simply plug in the speeds given. However, because the two objects
have the same initial speed, they will never collide unless they are heading for each other,
so one of the speed must be negative:
p i  p f
(3kg)( 2m / s )  (5kg)( 2m / s )  (8kg)(V f )
V f  .5m / s
11. A wooden block of mass M is moving at speed V in a straight line.
How fast would the bullet of mass m need to travel to stop the block (assuming that the
bullet became embedded inside)?
a.
mV
(m  M )
b.
Mv
(m  M )
c.
mV
M
d.
MV
m
e.
(m  M )V
m
*D. In order to stop the block, the total momentum of the bullet/block system must be
zero:
p  0
mV? 
V? 
MV
m
M(-V)  0
12.
(2004B, 75%)
An empty sled of mass M moves without friction across a frozen pond at speed Vo. Two
objects are dropped vertically onto the sled one at a time: first an object of mass m and
then an object of mass 2m. Afterward the sled moves with speed Vf. What would be the
final speed of the sled if the objects were dropped into it in reverse order?
a. Vf/3
b. Vf/2
c. Vf
e. 2Vf
e. 3 Vf.
*C
No matter the order of dropping, the sled is having an inelastic collision with a total mass
of 3m, so it has the same final speed either way.