Chapter 7
Problem 27
A skateboarder jumps on a moving skateboard from the side. Does the skateboardslow down or speed up in this process? Explain, using conservation ofmomentum.
Short Answer
Expert verified
Answer: The skateboard slows down when the skateboarder jumps on it from the side.
See the step by step solution
Step by step solution
TABLE OF CONTENTS :
TABLE OF CONTENTS
Step 1: Define the initial situation
Before the skateboarder jumps on the skateboard, the skateboard is moving with a certain velocity and mass. Let's define the mass of the skateboard as M_s and the initial velocity of the skateboard as V_s.
Step 2: Define the situation after the skateboarder jumps on the skateboard
When the skateboarder jumps on the skateboard, their mass will influence the movement of the skateboard. Let's define the mass of the skateboarder as M_r and their velocity during the jump as V_r.
Step 3: Calculate the initial momentum of the skateboard and skateboarder
The initial momentum of the skateboard can be calculated as the product of its mass (M_s) and its initial velocity (V_s). Similarly, the initial momentum of the skateboarder is the product of their mass (M_r) and their velocity (V_r):Initial momentum of skateboard (P_s) = M_s * V_s Initial momentum of skateboarder (P_r) = M_r * V_r
Step 4: Calculate the final momentum of the combined system
After the skateboarder jumps on the skateboard, they land and both the skateboarder and skateboard are moving together. The final momentum of the system is the sum of the momenta of the skateboard and skateboarder:Final momentum of system (P_f) = (M_s + M_r) * V_fwhere V_f is the final velocity of the combined system (skateboard and skateboarder).
Step 5: Apply the conservation of momentum principle
According to the conservation of momentum principle, the total initial momentum of the individual parts of the system (the skateboard and skateboarder) must equal the final momentum of the combined system:P_s + P_r = P_f
Step 6: Analyze whether the skateboard slows down or speeds up
To determine whether the skateboard slows down or speeds up when the skateboarder jumps on it, we can compare the initial velocity of the skateboard (V_s) to the final velocity of the combined system (V_f) by substituting the expressions for P_s, P_r, and P_f from Steps 3 and 4 into the conservation of momentum equation from Step 5:M_s * V_s + M_r * V_r = (M_s + M_r) * V_fNow, we can solve for V_f and analyze the expression to see how it compares to V_s.If V_f > V_s, then the skateboard speeds up.If V_f < V_s, then the skateboard slows down.Since the final momentum of the system will always be equal to the initial momentum, and the total mass of the system has increased (M_s + M_r), the final velocity of the combined system (V_f) must be less than the initial velocity of the skateboard (V_s). So, we can conclude that the skateboard slows down when the skateboarder jumps on it from the side.
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