A 65.0 kg ice skater moving to the right with a velocity of 2.47 m/s throws a 0.18 kg snowball to the right with a velocity of 24.3 m/s relative to the ground.
(a) What is the velocity of the ice skater after throwing the snowball? Disregard the friction between the skates and the ice.
_______ m/s to the right
(b) A second skater initially at rest with a mass of 62.00 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
_______ m/s to the right
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the event involving the ice skater and snowball is equal to the total momentum after the event.
(a) To find the velocity of the ice skater after throwing the snowball, we need to calculate the initial momentum of the system before the event, and then divide it by the ice skater's mass to find the final velocity.
The initial momentum before the event can be calculated using the formula: p_initial = m1 * v1 + m2 * v2, where m1 and m2 are the masses, and v1 and v2 are the velocities.
For the ice skater:
- mass (m1) = 65.0 kg
- velocity (v1) = 2.47 m/s
For the snowball:
- mass (m2) = 0.18 kg
- velocity (v2) = 24.3 m/s
Now, substitute these values into the formula:
p_initial = (65.0 kg) * (2.47 m/s) + (0.18 kg) * (24.3 m/s)
Calculating the initial momentum gives us:
p_initial = 160.55 kg·m/s
Since momentum is conserved, the total momentum after the event will also be 160.55 kg·m/s. Since the snowball is thrown in the same direction as the ice skater's initial velocity, the final momentum will only be due to the ice skater's mass and velocity.
Now, we can find the final velocity of the ice skater using the formula: p_final = m_final * v_final
Substituting the known values:
160.55 kg·m/s = (65.0 kg + 0.18 kg) * (v_final)
Simplifying the equation:
160.55 kg·m/s = 65.18 kg * (v_final)
Dividing both sides of the equation by 65.18 kg:
v_final = 2.46 m/s
Therefore, the velocity of the ice skater after throwing the snowball is 2.46 m/s to the right.
(b) To find the velocity of the second skater after catching the snowball in a perfectly inelastic collision, we can use the principle of conservation of momentum again.
The initial momentum before the collision is the momentum of the snowball and the ice skater:
p_initial = (65.0 kg) * (2.47 m/s) + (0.18 kg) * (24.3 m/s)
The total momentum after the collision is the momentum of the second skater:
p_final = (65.0 kg + 0.18 kg + 62.00 kg) * v_final_second_skater
Since the collision is perfectly inelastic, the snowball and second skater will stick together and move with the same final velocity.
Setting the initial and final momenta equal:
p_initial = p_final
Substituting the known values and velocity of the ice skater found in part (a):
(65.0 kg) * (2.47 m/s) + (0.18 kg) * (24.3 m/s) = (65.0 kg + 0.18 kg + 62.00 kg) * v_final_second_skater
Simplifying the equation:
160.55 kg·m/s = (127.18 kg) * v_final_second_skater
Dividing both sides of the equation by 127.18 kg:
v_final_second_skater = 1.26 m/s
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is 1.26 m/s to the right.