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.