Two ice skaters, Daniel (mass 70.0kg ) and Rebecca (mass 45.0kg ), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 14.0m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 6.00m/s at an angle of 53.1∘ from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink.
What is the magnitude of Daniel's velocity after the collision?
What is the direction of Daniel's velocity after the collision?
32. Two ice skaters stand at rest in the center of an ice rink. When they push off against each 45-kg skater acquires a speed of 0.62 m/s. If the speed of the other skater is 0.89 m/s, what is that skater's mass?
To find the magnitude and direction of Daniel's velocity after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
The total momentum of the system before the collision is equal to the total momentum after the collision. Mathematically:
(mass of Daniel × velocity of Daniel before collision) + (mass of Rebecca × velocity of Rebecca before collision) = (mass of Daniel × velocity of Daniel after collision) + (mass of Rebecca × velocity of Rebecca after collision)
2. Conservation of Kinetic Energy:
The total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision. Mathematically:
(1/2 × mass of Daniel × (velocity of Daniel before collision)^2) + (1/2 × mass of Rebecca × (velocity of Rebecca before collision)^2) = (1/2 × mass of Daniel × (velocity of Daniel after collision)^2) + (1/2 × mass of Rebecca × (velocity of Rebecca after collision)^2)
Now let's solve these equations step by step:
Step 1: Find the initial momentum of Daniel and Rebecca
Momentum is defined as mass × velocity. Therefore:
Momentum of Daniel before collision = mass of Daniel × velocity of Daniel before collision
Momentum of Rebecca before collision = mass of Rebecca × velocity of Rebecca before collision
Momentum of Daniel before collision = (70.0 kg) × (0 m/s) = 0 kg·m/s (since Daniel is at rest)
Momentum of Rebecca before collision = (45.0 kg) × (14.0 m/s) = 630 kg·m/s
Step 2: Find the final momentum of Daniel and Rebecca
From the conservation of momentum, we know that the total momentum before collision is equal to the total momentum after collision. Therefore:
(0 kg·m/s) + (630 kg·m/s) = (70.0 kg) × (velocity of Daniel after collision) + (45.0 kg) × (velocity of Rebecca after collision)
Step 3: Find the velocities of Daniel and Rebecca after collision
To solve for the velocities of Daniel and Rebecca after the collision, we need one more equation, which is the equation representing the angle between Rebecca's initial and final velocities. Let's call it θ.
Using vector addition, we know that Rebecca's initial velocity (14.0 m/s) and her final velocity (6.00 m/s) at an angle of 53.1° form a triangle. Using the Law of Cosines, we can write the equation:
(velocity of Rebecca after collision)^2 = (velocity of Rebecca before collision)^2 + (velocity of Daniel after collision)^2 - 2 × (velocity of Rebecca before collision) × (velocity of Daniel after collision) × cos(θ)
Substituting the known values:
(6.00 m/s)^2 = (14.0 m/s)^2 + (velocity of Daniel after collision)^2 - 2 × (14.0 m/s) × (velocity of Daniel after collision) × cos(53.1°)
Solve this equation to get the velocity of Daniel after the collision.
Step 4: Find the magnitude and direction of Daniel's velocity
Once you have the velocity of Daniel after the collision, you have the magnitude of the velocity. To find the direction, you can find the angle between Daniel's final velocity vector and a reference axis (e.g., the x-axis in this case) using the inverse tangent function. The angle can be found using the equation:
θ = tan^(-1)((velocity of Daniel after collision) / (velocity of Rebecca before collision))
Substituting the known values, solve this equation to get the direction of Daniel's velocity after the collision.