A 67.3 kg ice skater moving to the right with
a velocity of 2.52 m/s throws a 0.151 kg snowball to the right with a velocity of 33.7 m/s
relative to the ground.
What is the velocity of the ice skater after
throwing the snowball? Disregard the friction
between the skates and the ice.
Answer in units of m/s
A second skater initially at rest with a mass
of 61.4 kg catches the snowball.
What is the velocity of the second skater
after catching the snowball in a perfectly inelastic collision?
Answer in units of m/s
To solve these questions, we will apply the principle of conservation of momentum.
1. Velocity of the ice skater after throwing the snowball:
The momentum of an object is given by the product of its mass and velocity. So, to determine the velocity of the ice skater after throwing the snowball, we need to calculate the initial and final momentum of the system.
The initial momentum of the system (before throwing the snowball) is the sum of the momentum of the skater and the snowball:
Initial momentum = (mass of skater * velocity of skater) + (mass of snowball * velocity of snowball)
Given:
Mass of skater = 67.3 kg
Velocity of skater = 2.52 m/s
Mass of snowball = 0.151 kg
Velocity of snowball = 33.7 m/s
Initial momentum = (67.3 kg * 2.52 m/s) + (0.151 kg * 33.7 m/s)
Calculate the initial momentum by substituting the values:
Initial momentum = 169.8966 kg*m/s + 5.0947 kg*m/s
Initial momentum = 174.9913 kg*m/s
In a perfectly elastic collision, the total momentum remains constant. Hence, the final momentum of the system is also 174.9913 kg*m/s.
After throwing the snowball, the ice skater will be moving in the opposite direction to conserve momentum (since the snowball goes in the opposite direction).
Let's assume the velocity of the ice skater after throwing the snowball is v.
So, the final momentum of the system (after throwing the snowball) will be:
Final momentum = (mass of skater * velocity of skater) - (mass of snowball * velocity of snowball)
Substituting the values:
Final momentum = (67.3 kg * v) - (0.151 kg * 33.7 m/s)
174.9913 kg*m/s = 67.3 kg * v - 5.0947 kg*m/s
Rearranging the equation to solve for v:
v = (174.9913 kg*m/s + 5.0947 kg*m/s) / 67.3 kg
Calculate the velocity by substituting the values:
v = 2.6095 m/s
Therefore, the velocity of the ice skater after throwing the snowball is 2.6095 m/s.
2. Velocity of the second skater after catching the snowball:
Since this is a perfectly inelastic collision, the two skaters will stick together after the collision. To determine the final velocity, we need to apply the principle of conservation of momentum again.
The momentum before the collision is the sum of the momentum of the initially moving skater and the momentum of the snowball:
Initial momentum = (mass of skater * velocity of skater) + (mass of snowball * velocity of snowball)
Given:
Mass of skater = 61.4 kg (second skater, initially at rest)
Mass of snowball = 0.151 kg
Velocity of skater = 2.6095 m/s (calculated in the previous question)
Velocity of snowball = 33.7 m/s
Initial momentum = (61.4 kg * 0 m/s) + (0.151 kg * 33.7 m/s)
Calculate the initial momentum by substituting the values:
Initial momentum = 0 kg*m/s + 5.0947 kg*m/s
Initial momentum = 5.0947 kg*m/s
In a perfectly inelastic collision, the total momentum is conserved. So, the final momentum of the system will also be 5.0947 kg*m/s.
The final momentum of the system (after catching the snowball) is given by the sum of the momentum of the combined skaters:
Final momentum = (mass of skater 1 + mass of skater 2) * final velocity
Let the final velocity of both skaters be V.
Substituting the values:
5.0947 kg*m/s = (67.3 kg + 61.4 kg) * V
Simplifying the equation:
5.0947 kg*m/s = 128.7 kg * V
Solving for V:
V = 5.0947 kg*m/s / 128.7 kg
Calculate the velocity by substituting the values:
V ā 0.039676 m/s
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 0.039676 m/s.