21. Two skaters on a frictionless icy surface push apart from one another. One skater has a mass Mmuch greater than the mass m of the second skater. After some time the two skaters are at adistance d apart. How far has the lighter skater moved from her original position?
Question
- Two skaters on a frictionless icy surface push apart from one another. One skater has a mass Mmuch greater than the mass m of the second skater. After some time the two skaters are at adistance d apart. How far has the lighter skater moved from her original position?
Solution
This problem can be solved using the principle of conservation of momentum.
Step 1: Understand the principle of conservation of momentum The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. Momentum is the product of mass and velocity of an object.
Step 2: Apply the principle to the problem Before the skaters push off from each other, their total momentum is zero because they are stationary. After they push off, the total momentum should still be zero because there are no external forces.
Let's denote the distance the lighter skater moved as d1 and the distance the heavier skater moved as d2. The velocity of the lighter skater is d1/t and the velocity of the heavier skater is d2/t, where t is the time it takes for them to be d apart.
Step 3: Set up the equation According to the conservation of momentum, the momentum of the lighter skater should be equal and opposite to the momentum of the heavier skater. So, we have:
m*(d1/t) = M*(d2/t)
Step 4: Solve the equation We know that d1 + d2 = d, so d2 = d - d1. Substitute d2 into the equation, we get:
m*(d1/t) = M*((d - d1)/t)
Solving this equation for d1, we get:
d1 = d*M/(m+M)
So, the lighter skater has moved d*M/(m+M) from her original position.
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