A 50-kg woman and a 25-kg boy stand motionless facing each other on friction-free roller blades. The woman pushes the boy, who moves away at a speed of 5.0 m/s. What is the woman’s speed in the opposite direction?
Here's a clue (from the law of cnservation of momentum)
Their momenta are equal and opposite, so that the total momentum remains zero.
M1*V1 = -M2*V2
V2/V1 = ?
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the woman pushes the boy is equal to the total momentum after the push.
The momentum of an object is given by the product of its mass and velocity:
Momentum = Mass × Velocity
Let's assign variables to the given quantities:
- Woman's mass = 50 kg
- Boy's mass = 25 kg
- Boy's initial velocity = 0 m/s (since he is initially motionless)
- Boy's final velocity = 5.0 m/s (after being pushed)
- Woman's final velocity = ? (what we need to find)
The total momentum before the push is zero because both the woman and the boy are motionless.
Total momentum before = Woman's momentum before + Boy's momentum before
Total momentum before = Woman's mass × Woman's initial velocity + Boy's mass × Boy's initial velocity
Total momentum before = 50 kg × 0 m/s + 25 kg × 0 m/s
Total momentum before = 0 kg∙m/s
The total momentum after the push is the sum of the woman's and the boy's momentum after. The woman moves in the opposite direction, so her velocity will be negative:
Total momentum after = Woman's momentum after + Boy's momentum after
Total momentum after = Woman's mass × Woman's final velocity + Boy's mass × Boy's final velocity
Total momentum after = 50 kg × (-Woman's final velocity) + 25 kg × 5.0 m/s
Total momentum after = -50 kg × Woman's final velocity + 125 kg∙m/s
According to the principle of conservation of momentum:
Total momentum before = Total momentum after
0 kg∙m/s = -50 kg × Woman's final velocity + 125 kg∙m/s
Now, solve for "Woman's final velocity":
50 kg × Woman's final velocity = 125 kg∙m/s
Woman's final velocity = 125 kg∙m/s / 50 kg
Woman's final velocity = 2.5 m/s
Therefore, the woman's speed in the opposite direction is 2.5 m/s.