Ian is firmly attached to a snowboard which is on a flat surface of ice, which you can assume to be frictionless. He and the snowboard are initially at rest. He then throws a 138 g ball which travels at a speed of 19.2 m/s, and finds himself moving backwards at a speed of 0.05 m/s. What is the combined mass of Ian and his snowboard (in kg)?
To solve this problem, we need to apply 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, assuming there are no external forces.
Let's denote the mass of the snowboard as M, and the mass of Ian as m.
Initially, the snowboard and Ian are at rest, so their total momentum is zero. The momentum of the ball is given by the product of its mass (m_ball = 138 g = 0.138 kg) and its velocity (v_ball = 19.2 m/s):
Momentum of the ball = m_ball * v_ball = 0.138 kg * 19.2 m/s = 2.6556 kg·m/s.
When Ian throws the ball, he experiences an equal and opposite change in momentum due to the law of action and reaction. Since he moves backward, his momentum is negative. In this case, we'll denote Ian's velocity as v_ian = -0.05 m/s.
So, the momentum of Ian and the snowboard after the throw is the sum of their individual momenta, which must be equal in magnitude but opposite in sign to the momentum of the ball:
Total momentum after = Momentum of Ian + Momentum of the snowboard
0 = m * v_ian + M * v_ian
Since both Ian and the snowboard are moving backward, their velocities have the same magnitude but opposite signs. Therefore, we can rewrite the equation as:
0 = (m + M) * v_ian.
Now, we can solve for the combined mass (m + M):
(m + M) * v_ian = 0.
But we know that v_ian ≠ 0, so we conclude that (m + M) = 0.
Hence, the combined mass of Ian and the snowboard is 0 kg.