A 74.0 kg ice skater moving to the right with a velocity of 2.90 m/s throws a 0.15 kg snowball to the right with a velocity of 23.2 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.
(b) A second skater initially at rest with a mass of 61.50 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
To solve this problem, we can apply the law of conservation of momentum. According to this law, the initial momentum of a system is equal to the final momentum of the system, assuming there are no external forces acting on it.
(a) First, let's find the momentum of the ice skater before throwing the snowball.
Momentum (p) is calculated by multiplying mass (m) by velocity (v).
The momentum of the ice skater before throwing the snowball is given by:
p1 = m1 * v1
Given:
m1 = 74.0 kg (mass of the ice skater)
v1 = 2.90 m/s (velocity of the ice skater)
Substituting the given values, we get:
p1 = 74.0 kg * 2.90 m/s
p1 = 214.60 kg·m/s
Now, let's find the momentum of the snowball.
The momentum of the snowball is given by:
p2 = m2 * v2
Given:
m2 = 0.15 kg (mass of the snowball)
v2 = 23.2 m/s (velocity of the snowball)
Substituting the given values, we get:
p2 = 0.15 kg * 23.2 m/s
p2 = 3.48 kg·m/s
Since there are no external forces acting on the system (disregarding friction), the total momentum before throwing the snowball is equal to the total momentum after throwing it.
Therefore, the velocity of the ice skater after throwing the snowball can be determined by dividing the sum of the individual momenta by the total mass.
Total mass (m_total) = m1 + m2
m_total = 74.0 kg + 0.15 kg
m_total = 74.15 kg
Total momentum after throwing the snowball (p_total) = p1 + p2
p_total = 214.60 kg·m/s + 3.48 kg·m/s
p_total = 218.08 kg·m/s
Velocity of the ice skater after throwing the snowball (v_final) = p_total / m_total
v_final = 218.08 kg·m/s / 74.15 kg
v_final ≈ 2.942 m/s
Therefore, the velocity of the ice skater after throwing the snowball is approximately 2.942 m/s to the right.
(b) To find the velocity of the second skater after catching the snowball, we can use the equation for conservation of momentum again. In a perfectly inelastic collision, the two objects stick together and move with a common final velocity.
Let's denote the velocity of the second skater after catching the snowball as v_final2.
Total mass after the collision (m_total_final) = m1 + m2 + m_skater2
m_total_final = 74.0 kg + 0.15 kg + 61.50 kg
m_total_final = 135.65 kg
Total momentum after the collision (p_total_final) = m_total_final * v_final2
Since the snowball and the second skater are initially at rest, their momenta are zero before the collision. Thus, the momentum of the system is simply the initial momentum of the first skater before throwing the snowball.
Therefore, p_total_final = p1
Substituting the known values, we have:
m_total_final * v_final2 = 214.60 kg·m/s
Solving for v_final2, we get:
v_final2 = p1 / m_total_final
v_final2 = 214.60 kg·m/s / 135.65 kg
v_final2 ≈ 1.584 m/s
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 1.584 m/s to the right.