Momentum, Collisions and Explosions Legacy Problem #10 Guided Solution
Problem*
Cassie has just finished her session on the trampoline during PE. As she prepares to exit the trampoline, her vertical momentum is reduced by a series of three resistive impulses with the bounce mat. Just prior to this series of impulses, her 48.5-kg body is moving downward at 8.20 m/s. On the first impulse, Cassie experiences an average upward force of 230 N for 0.65 seconds. The second impulse of 112 N•s lasts for 0.41 seconds. The last impulse involves an average upward force of 116 N which cases a 84 kg•m/s momentum change. What vertical velocity does Cassie have after these three impulses?
Audio Guided Solution
This is a difficult problem which involves the analysis of a motion in which are a series of three impulses acting upon a student in gym class in order to change the vertical momentum of the student over the course of three collisions. Now we're told the mass of the student and the original velocity of the student as she approaches her first impulse. We can find the original momentum by simply multiplying 48.5 kilograms times 8.20 meters per second. Once we get this original momentum we should write it down and we should recognize that this momentum is going to change. It's going to be reduced three times and if we can find either the impulse or the momentum change for each of the three individual impulses we should be able to calculate the final momentum and if we get that we can divide by mass and find the final velocity. So the way I'm going to approach this is I'm going to treat it one collision at a time beginning with a momentum of 397.7 kilograms times meter per second of momentum. The student encounters her first impulse. It's an impulse that has resulted from 230 newtons of force acting for 0.65 seconds. If I multiply the F times the T I get 149.5 newtons times second impulse. It's a resistive impulse so it takes away momentum. So what I'm going to do is subtract 149.5 units of momentum from the original amount 397.7. This gives me a new momentum after the first impulse. I should write it down and then I'm going to treat the next part of the problem in which there's a second impulse of 112 units. The fact that that second impulse endures for 0.41 seconds is of little importance here. What is of importance is the fact that it's a 112 unit momentum change. So what I'm going to do is subtract 112 units of momentum from what I had after the first impulse which was 248.2. Now I subtract 112 and I get 136.2 units of momentum going into my third impulse. And then my third impulse there's an 84 kilogram times meter per second momentum change that takes place. The fact that the force was 116 newtons really doesn't matter. What matters is how much momentum change there is. So I've got an 84 unit momentum change. It's a resistive impulse once more so it takes away momentum. So I'm going to subtract the 84 from the 136.2 and I get 52.2 units of momentum. That's the final momentum of Cassie after she encounters this series of three resistive impulses. Now I can use the mass of Cassie in order to determine the final velocity. So I take this momentum and I divide by the mass of 48.5 and I get a value of 1.076 meters per second. I can round that to 1.08 which would be three significant digits.
Solution
1.08 m/s
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(m = 1.50 \unit{kg}\), \(v_i = 2.68 \unit{\meter\per\second}\), \(F = 4.98 \unit{\newton}\), \(t = 0.133 \unit{\second}\), \(v_f = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
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