Question

Two skaters one traveling north and one traveling 20 degrees north of west collide in the middle of an ice rink. The one traveling north has a mass of 50.0 kg and is traveling at 6.00 m/s, the one traveling north of west has a mass of 60.0 kg and is traveling at 4.00 m/s,

a) What is their velocity right after collision if they latch on to each other? (give your answer as a magnitude and an angle)

b) If the radius of the rink is 20.0 m how long will it take them to hit the wall?

Answer #1

First notice that this is a conservation of linear momentum problem. The momentum p of an object (or skater) is given by , where m is their mass and v is their speed. By conservation of linear momentum, their sum of 2 skaters individual momenta is equal to their momentum (attached to each other) after the collision. Be careful to note that momentum is a vector quantity and therefore has a direction component.

**(a)**

**Step 1)** Find the momentum before the collision
of skater 1, who is travelling due north. Her mass is
, and her velocity is
. Since , her momentum
is:

Skater 1's momentum is in the positive y-direction.

**Step 2)** Find skater 2's (who is moving north of
west) initial momentum. His mass is
and velocity is
, so his momentum is:

**Step 3)** Next find the momentum of the pair of
skaters stuck together after the collision. Their combined mass is
, and their speed is . Their momentum
is:

**Step 4)** Now use conservation of linear momentum
to find their final velocity after the collision. First Use
conservation of momentum in the x-direction. Add the x-components
of momentum from the each skater's initial momentum and set it
equal to the x-component of the final momentum.

The final speed in the negative x-direction is 2.05 meters per second.

**Step 5)** Find the y-component of the final speed
by adding together the y-components of the skater's initial momenta
and set it equal to the final momentum in the y-direction.

Their final speed in the positive y-direction is 3.47 meters per second.

**Step 6)** To find the magnitude of the velocity,
use the formula
where
and
.

The magnitude of the velocity is 4.03 meters per second.

**Step 7)** To find the direction, recall that
. Plug in
and
and solve for the angle.

The direction of the skaters after the collision is 59.4 degrees.

**(b)**

**Step 8)** To determine how long they have before
they hit the wall, recall that the definition of speed is
where r is the distance and t is the time. The speed was found to
be
and the distance was given to be .
Plug these values into
and solve for t.

The skaters will hit the wall in 4.96 seconds.

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collides with him. After the collision, Rebecca has a velocity of
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