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A man running on a road had a momentum of 360 kilograms meters per second.
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The road ahead of the man was covered by sand.
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The man ran through the sand for 12 seconds and his momentum when he had crossed the sand-covered part of the road was 330 kilograms meters per second.
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What average force did the sand apply to the man while he ran across it?
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Okay, so to answer this question, it might be useful to first visualize what’s going on in the scenario.
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So here’s a diagram of our man running on the road.
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Now, we’ve been told that initially the man had a momentum of 360 kilograms meters per second.
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But then, something happened that changed the man’s momentum.
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Specifically, the man started running through some sand that was on the road and the sand on the road exerted a force on the man.
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And this force was such that the man’s momentum after he crossed the sand-covered part of the road was 330 kilograms meters per second.
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In other words, he’d lost momentum.
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Now, what we need to do is to work out the average force exerted by the sand onto the man while he ran across it.
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And we need to do this having been given information about his initial momentum, his final momentum, and how long he ran across the sand for.
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To do this, we need to recall that an impulse is defined as the force applied on an object multiplied by the time interval Δ𝑡 for which that force is applied on the object.
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And with this, we can also recall that an impulse is equal to the change in momentum of that object.
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And we’ll call the change in momentum Δ𝑃.
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So in other words, the force applied on our object multiplied by the amount of time for which that force is applied is equal to the change in momentum of that object.
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And we can use this information because we’ve been given the initial and final momentum.
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So we can work out the change in momentum.
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And as well as this, we know the amount of time for which the force is applied because we know that the man ran across the sand for 12 seconds.
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So we can rearrange this equation to find out what the force applied on the man by the sand was.
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To do this, we divide both sides of the equation by Δ𝑡.
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This leaves us with the force on the left-hand side 𝐹 and on the right we’ve got Δ𝑃, the change in momentum, divided by Δ𝑡, the amount of time for which the force is applied.
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And at this point, we can substitute in the values.
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So firstly, what’s the change in momentum?
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Well, the change in momentum is simply equal to the final momentum, which we’ve called 𝑃 sub final, minus the initial momentum, which we’ve called 𝑃 sub initial.
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Now, the final momentum happens to be 330 kilograms meters per second and the initial momentum happens to be 360 kilograms meters per second.
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So this expression is going to give us the change in momentum Δ𝑃.
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And we can’t forget to divide this by the time interval for which the force is applied, which we’ve been told is 12 seconds, because as we said earlier the man runs across the sand for 12 seconds.
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And hence, the sand applies a force for 12 seconds on the man.
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At this point, we can evaluate the right-hand side of this equation.
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And so we find that the force applied on the man by the sand is negative 2.5 newtons.
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And this makes sense because before the man runs to the sand, he’s got a momentum of 360 kilograms meters per second.
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Afterwards, he’s got a momentum of 330 kilograms meters per second.
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In other words, he’s lost momentum or another way to put it is that the momentum that he has towards the right is less than what it was before.
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And the way that the sand does this is by exerting a force on the man to the left — in other words, in the negative direction.
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Hence, we have a negative value for the force.
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And so we have our final answer.
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The average force applied by the sand on the man is negative 2.5 newtons.