A 73.5 kg ice skater moving to the right with a velocity of 2.64 m/s throws a 0.18 kg snowball to the right with a velocity of 23.8 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 64.00 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
(a) To find the velocity of the ice skater after throwing the snowball, we can use the law of conservation of momentum. According to this law, the total momentum before the snowball is thrown should be equal to the total momentum after the snowball is thrown.
The momentum of an object is calculated by multiplying its mass by its velocity. So we can calculate the initial momentum of the ice skater as:
Initial momentum of ice skater = mass of ice skater × velocity of ice skater
Given that the mass of the ice skater is 73.5 kg and the velocity of the ice skater is 2.64 m/s, we can calculate the initial momentum as:
Initial momentum of ice skater = 73.5 kg × 2.64 m/s = 193.44 kg·m/s
Now, let's calculate the momentum of the thrown snowball. The momentum of the snowball can be calculated in the same way:
Momentum of snowball = mass of snowball × velocity of snowball
Given that the mass of the snowball is 0.18 kg and the velocity of the snowball is 23.8 m/s, we can calculate the momentum as:
Momentum of snowball = 0.18 kg × 23.8 m/s = 4.284 kg·m/s
According to the law of conservation of momentum, the total momentum after the snowball is thrown should be equal to the initial momentum of the ice skater. So we can write the following equation:
Initial momentum of ice skater = momentum of snowball + momentum of ice skater after throwing
Plugging in the known values, we get:
193.44 kg·m/s = 4.284 kg·m/s + momentum of ice skater after throwing
To find the momentum of the ice skater after throwing, we rearrange the equation:
momentum of ice skater after throwing = 193.44 kg·m/s - 4.284 kg·m/s = 189.156 kg·m/s
Finally, we can find the velocity of the ice skater after throwing by dividing the momentum by the mass of the ice skater:
velocity of ice skater after throwing = momentum of ice skater after throwing / mass of ice skater
Plugging in the values, we get:
velocity of ice skater after throwing = 189.156 kg·m/s / 73.5 kg ≈ 2.575 m/s (rounded to three significant figures)
Therefore, the velocity of the ice skater after throwing the snowball is approximately 2.575 m/s.
(b) In this case, the snowball is caught by a second skater, and the collision is perfectly inelastic. In a perfectly inelastic collision, the two objects stick together after the collision, forming a single object.
To find the velocity of the second skater after catching the snowball, we can also use the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
Again, we calculate the initial momentum of the snowball and the second skater. The initial momentum of the snowball remains the same as calculated in part (a): 4.284 kg·m/s.
The initial momentum of the second skater can be calculated as:
Initial momentum of second skater = mass of second skater × velocity of second skater
Given that the mass of the second skater is 64.00 kg and initially at rest, the initial momentum of the second skater is 0.
According to the law of conservation of momentum, the total momentum after the collision should be equal to the initial momentum of the snowball and the second skater. So we can write the following equation:
Initial momentum of snowball + Initial momentum of second skater = Total momentum after the collision
Plugging in the known values, we get:
4.284 kg·m/s + 0 = Total momentum after the collision
Simplifying the equation, we find that the total momentum after the collision is also 4.284 kg·m/s.
Since the snowball and the second skater stick together, their total mass after the collision is the sum of their masses:
Total mass after collision = mass of snowball + mass of second skater
Total mass after collision = 0.18 kg + 64.00 kg = 64.18 kg
To find the velocity of the second skater after catching the snowball, we divide the total momentum after the collision by the total mass after the collision:
Velocity of second skater after collision = Total momentum after collision / Total mass after collision
Plugging in the values, we get:
Velocity of second skater after collision = 4.284 kg·m/s / 64.18 kg ≈ 0.067 m/s (rounded to three significant figures)
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 0.067 m/s.