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A child of mass 30 kilograms slides down a slide.
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Near the top of the slide, the child’s velocity is 0.65 meters per second, while near the bottom of the slide, the child’s velocity is 1.35 meters per second.
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How much does the child’s momentum change between these two positions?
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In this question, we have a child who’s sliding down a slide.
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Let’s begin by drawing ourselves a diagram to show the situation.
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So here’s a picture of our slide, and we’re asked to consider two different positions of the child on this slide.
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The first position is near the top of the slide.
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We’re told that at this position the velocity of the child is 0.65 meters per second.
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We’ll call this velocity 𝑣 subscript one.
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And we know that its direction will be down the slide.
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So we have that 𝑣 subscript one has a magnitude of 0.65 meters per second.
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The second position that we are asked to consider is near the bottom of the slide.
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At this position, we’re told that the velocity of the child is 1.35 meters per second.
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We’ll call this velocity 𝑣 subscript two.
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And again, we know that the direction is down the slide.
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We have that the magnitude of 𝑣 two is equal to 1.35 meters per second.
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We are asked to work out the change in the child’s momentum between these two positions.
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Recall that the momentum of an object is equal to the product of the object’s mass and the object’s velocity.
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Mathematically, we can write this as 𝑝 is equal to 𝑚 times 𝑣, where 𝑝 is the object’s momentum, 𝑚 is the object’s mass, and 𝑣 is the object’s velocity.
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Now, we know the velocities of the child at each of the two positions.
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And we also know the child’s mass because we’re told in the question that the child has a mass of 30 kilograms.
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If we label this mass as 𝑚, then we can say that 𝑚 is equal to 30 kilograms.
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So we know the value of 𝑚.
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And we know the value of 𝑣 at each of the two positions.
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So we have all of the information that we need to calculate the momentum 𝑝.
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Looking again at the question, we see that we’re not just asked to find the momentum of the child.
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We’re asked to find the change in their momentum between the two positions.
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If we label the child’s momentum near the top of the slide as 𝑝 one and the child’s momentum near the bottom of the slide as 𝑝 two, then we have that the change in momentum between the two positions, which we’ll call 𝛥𝑝, is equal to 𝑝 two minus 𝑝 one.
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In other words, to find the change in momentum from near the top of the slide to near the bottom, we need to calculate the momentum near the top of the slide.
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So that’s the value of 𝑝 one.
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And we need to subtract this from the value of the momentum near the bottom of the slide.
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So that’s 𝑝 two.
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Let’s begin by calculating the value of 𝑝 one.
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Since momentum 𝑝 is equal to mass 𝑚 multiplied by velocity 𝑣, we have that 𝑝 one, the momentum near the top of the slide, is equal to 𝑚, the mass of the child, multiplied by 𝑣 one, the velocity near the top of the slide.
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Now we can substitute into this equation that 𝑚 is equal to 30 kilograms and 𝑣 one is equal to 0.65 meters per second.
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This gives us that 𝑝 one is equal to 30 kilograms multiplied by 0.65 meters per second.
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Doing the multiplication, we get a result for the momentum 𝑝 one of 19.5 kilogram-meters per second.
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Now, let’s calculate the value of the momentum 𝑝 two.
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So that’s the momentum near the bottom of the slide.
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We have that 𝑝 two is equal to the mass 𝑚 multiplied by the velocity 𝑣 two.
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We know that the mass 𝑚 is equal to 30 kilograms and the value of 𝑣 two is 1.35 meters per second.
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Substituting these values into our expression for 𝑝 two, we get that 𝑝 two is equal to 30 kilograms multiplied by 1.35 meters per second.
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Then doing the multiplication, we get a result for 𝑝 two of 40.5 kilogram-meters per second.
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It’s worth pointing out that the direction of 𝑝 one and 𝑝 two will be the same as the directions of 𝑣 one and 𝑣 two.
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That is, both 𝑝 one and 𝑝 two are directed down the slide.
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The final step in answering this question is to take our values of 𝑝 one and 𝑝 two and substitute them into this equation here to calculate the value of 𝛥𝑝.
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So that’s the change in momentum of the child between near the top of the slide and near the bottom.
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Substituting these values, we get that 𝛥𝑝 is equal to 𝑝 two, which is 40.5 kilogram-meters per second, minus 𝑝 one, which is 19.5 kilogram-meters per second.
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Evaluating this expression gives us our final result for 𝛥𝑝.
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And this result is our answer to the question.
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And so we have found that the amount by which the child’s momentum changes between the two positions is 21 kilogram-meters per second.