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Planar Momentum

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Example Momentum Problems

EXAMPLE MOMENTUM PROBLEMS

1. Two masses are on a frictionless horizontal surface. The 65 kg mass is at rest, when struck by the 45 kg mass moving at 13.0 m/s. After the collision, the 45 kg mass has a velocity of magnitude 8.00 m/s at an angle of 53.1 degrees from its initial direction.

a) What is the magnitude of the 65 kg mass's velocity after the collision?
b) What is the direction of the 65 kg mass's velocity after the collision? (degrees from the 45 kg mass's original)
c) What is the change in total kinetic energy of the two masses as a result of the collision?

2. A skater with a mass of 80.0 kg is traveling due east at 4.00 m/s when she collides with another skater of mass 55.0 kg heading due south at 16.0 m/s. If they stay tangled together, what is their final velocity?

3. A woman and a man simultaneously dive from a 120 kg raft that is initially at rest. The woman (60 kg) jumps from the raft with a horizontal speed of 1.9 m/s due south, while the man (77 kg) jumps with a horizontal speed of 1.5 m/s due west. Calculate the magnitude and direction of the raft’s velocity immediately after their dives.

ANSWERS TO EXAMPLE MOMENTUM PROBLEMS

Momentum before the collision and the momenta of the two skaters, parting at right angles, after the collision are illustrated in this diagram.

1. This diagram illustrates the total momentum before the collision and the momenta of the two masses after the collision.

Total momentum is conserved.  This diagram illustrates the sum of the momentum vectors after skaters collide is the total momentum of the skaters before the collision.

This diagram illustrates conservation of momentum, showing the total momentum before the collision equal to the sum of the momenta of the skaters after the collision.

The momentum of each mass is related to the initial momentum by the sine of the opposite angle. Once you have the momentum of a mass, you can find velocity because you were given the mass.

Knowing the mass and momentum of each mass, you can now calculate the kinetic energy of each, and finally the change in kinetic energy from the 45 kg mass's original kinetic energy.

2.

Total momentum is a vector sum.

The initial momenta and total momentum are illustrated to the left.

p1 = (80)(4.00)

p2 = (55.0)(16.0)

pTOT = √(p12 + p22 )

ø = tan-1( p1 / p2)

After the collision pTOT is the same as above.

Since pTOT = (80 +55)v , solve for v, the only unknown.

To cancel the two components, the resultant must be an equilibrant.

3. Immediately after diving, the momenta of the woman is 114 kg-m/s [S] and of the man is 115.5 kg-m/s [W].

Since the total momentum before the event was zero, the momenta of the man, woman and raft after the event must add to zero.

ø = tan-1( 114/115.5) = 44.6º

The momentum of the raft after the event is given by

√(1142 + 115.52) = 162 kg-m/s

Since v = p/m, v = (162 kg-m/s) / (120 kg) = 1.35 m/s

The velocity of the raft is 1.35 m/s 135.4º away from the velocity of the man.

Momentum Problems

(See bottom of page for answers.)

A. COLLISIONS WITHOUT COUPLING

1. A pendulum consisting of a 0.8 kg ball attached to a string of length 2.3 m is released from rest when the string makes an angle of 53° with the vertical. At the bottom of its swing, the ball collides elastically with a block of mass m at rest on a horizontal frictionless surface. After the collision, the maximum angle the string makes with the vertical is 5.73°. What is the mass of the block?

B. COLLISIONS WITH COUPLING

1. Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity 14.5 m/s toward the east and the other is traveling north with speed v2. The vehicles collide in the intersection and stick together, traveling at an angle of 52.0° north of east. What was the initial speed of the northward-moving vehicle?

2. A 12 g bullet traveling at 500 m/s strikes an 0.8 kg block of wood that is balanced on a table edge 0.8 m above ground. If the bullet buries itself in the block, find the distance D at which the block hits the floor.

3. A 1.8-kg block, initially at rest, slides down a frictionless ramp that is angled at 35 degrees to the horizontal. At a point 0.45 m down the slope it collides with and sticks to a stationary block of mass 1.1 kg. The blocks then continue another 0.88 m down the ramp. How long does the whole event take?

C. EXPLOSIONS

1. A grenade thrown horizontally from a cliff explodes before hitting the ground. Neglecting air resistance, the center of mass of the grenade fragments, just after the explosion is ( zero , along parabolic path, moves horizontally, moves vertically ).

2. A block resting on a smooth surface explodes into three pieces. A 200-g piece travels at 1.4 m/s, a 300-g piece travels 0.90 m/s, and a third piece flies off at a speed of 1.8 m/s. If the angle between the first two pieces is 80 degrees, calculate the mass and direction of the third piece.

ANSWERS to MOMENTUM PROBLEMS

For solutions to all the problems on this page click here.

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