a 65-kg ice skater moving to the right with a velocity of 2.50m/s throws a .150kg snowball to the right with a velocity of 32.0m/s relative to the ground
a) what is the velocity of the ice skater after throwing the snowball?
b) a second skater initially at rest with a mass of 60kg catches the snowball. What is the velocity of the second skater after catching the snowball?
I’m concerned what u mean by adding the letter “u” in this equation
3.9
a) Well, if we take into account the conservation of momentum, the ice skater and the snowball must have equal but opposite momenta before and after the throw. Since the snowball is moving to the right with a velocity of 32.0m/s relative to the ground, our ice skater will experience an equal but opposite velocity change. So, after the throw, the ice skater's velocity will be -29.5m/s (still to the right, just a lot slower).
b) Now, let's consider the conservation of momentum again. The total momentum of the system before the catch is zero, since both skaters are initially at rest. After the catch, the momentum must still be zero. So, our second skater with a mass of 60kg will acquire a velocity of 0m/s to keep the momentum balanced. In other words, the poor skater won't be moving at all after catching the snowball.
To solve these problems, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, provided no external forces act on the system.
Let's break down the problems step by step:
a) What is the velocity of the ice skater after throwing the snowball?
The momentum of an object is given by its mass multiplied by its velocity. We can calculate the initial momentum of the ice skater and the snowball:
Initial momentum of the ice skater = mass of the ice skater × velocity of the ice skater
= (65 kg) × (2.50 m/s) = 162.5 kg·m/s (to the right)
Initial momentum of the snowball = mass of the snowball × velocity of the snowball
= (0.150 kg) × (32.0 m/s) = 4.8 kg·m/s (to the right)
Since no external forces act on the system after the snowball is thrown, the total momentum before throwing the snowball is equal to the total momentum after throwing the snowball. Therefore, the initial momentum of the ice skater plus the initial momentum of the snowball is equal to the final momentum of the ice skater:
Final momentum of the ice skater = Initial momentum of the ice skater + Initial momentum of the snowball
= 162.5 kg·m/s + 4.8 kg·m/s
= 167.3 kg·m/s (to the right)
The velocity of the ice skater after throwing the snowball can be calculated using the final momentum and the mass of the ice skater:
Final momentum of the ice skater = mass of the ice skater × velocity of the ice skater (after throwing the snowball)
167.3 kg·m/s = (65 kg) × (velocity of the ice skater after throwing the snowball)
Dividing both sides of the equation by the mass of the ice skater (65 kg):
(velocity of the ice skater after throwing the snowball) = 167.3 kg·m/s / 65 kg
The velocity of the ice skater after throwing the snowball is approximately 2.57 m/s to the right.
b) What is the velocity of the second skater after catching the snowball?
In this scenario, we have two skaters, one moving initially with a momentum and another at rest. The second skater catches the snowball without any external forces. We can apply the same principle of conservation of momentum.
The initial momentum of the system (the first skater and the snowball) before the snowball is caught by the second skater is equal to the final momentum of the system (both skaters):
Initial momentum of the system = Final momentum of the system
The initial momentum of the system is the initial momentum of the ice skater plus the initial momentum of the snowball:
Initial momentum of the system = Initial momentum of the ice skater + Initial momentum of the snowball
The final momentum of the system is the sum of the final momentum of the first skater (after throwing the snowball) and the final momentum of the second skater (after catching the snowball):
Final momentum of the system = Final momentum of the first skater + Final momentum of the second skater
Since the first skater continues moving to the right with a velocity of 2.57 m/s (as calculated in part a), the final momentum of the first skater is:
Final momentum of the first skater = mass of the first skater × velocity of the first skater (after throwing the snowball)
= (65 kg) × (2.57 m/s) = 166.9 kg·m/s (to the right)
The initial momentum of the system is equal to the final momentum of the system, so:
Initial momentum of the ice skater + Initial momentum of the snowball = Final momentum of the first skater + Final momentum of the second skater
(162.5 kg·m/s to the right) + (4.8 kg·m/s to the right) = 166.9 kg·m/s (to the right) + Final momentum of the second skater
Finally, to solve for the final momentum of the second skater:
Final momentum of the second skater = (162.5 kg·m/s to the right) + (4.8 kg·m/s to the right) - 166.9 kg·m/s (to the right)
= 0.4 kg·m/s to the left
The velocity of the second skater after catching the snowball is approximately 0.0067 m/s to the left.
m1=65 kg, m2 =0.15 kg, m3 = 60 kg,
v1 = 2.5 m/s, v2 = 32 m/s.
(m1 + m2)•v1 =m2•v2 + m1•u,
u = {(m1+m2) •v1 – m2•v2}/m1 = {65.15•2.5 -0.15•32}/65 = 2.43 m/s,
m2•v2 = (m2+m3) •u1,
u1 = m2•v2/(m2+m3) = 0.15•32/(60+0.15) =
= 0.08 m/s.