Applying ideas about motion......

[Jimmy87]

Homework Statement

Long answer question. Simon is a long jumper. He tries to run as fast as he can before he jumps. This enables him to jump much further than if he did without running.

Question: Use ideas about forces and motion to explain how running helps Simon jump further

Homework Equations

KE = 1/2mv^2
GPE = mgh
P = mv

The Attempt at a Solution

The forces acting during the run are static friction on the track's surface. As the runner pushes on the track the frictional force exerts an equal and opposite force on the runner. If this force is bigger than the resistive force, the runner will accelerate. This will cause the kinetic energy to increase.

The distance of the jump depends on the time the jumper is in the air and the forward speed during this time in the air. The time in the air is determined by the vertical velocity which is determined by the size of the vertically downward reaction force form the floor before he jumps. So if he pushes vertically down on the floor before he jumps he will have a vertical velocity and the larger this force is the larger the velocity. A larger vertical velocity gives a greater vertical kinetic energy which means there is a larger gravitational potential energy which means a greater distance (height) which means a longer time in the air. Is it ok to say that if you have more vertical KE then you are in the air longer because the opposing gravitational force is constant so if you have more kinetic energy you will take longer to stop (work = force x distance) before you come back down to the ground?

What I'm confused about is I found a similar question to this and the answer said the following:

"The distance of the jump is linked to the time in the air which is why the running is important beforehand"

I am confused with this as I would have thought that the time in the air is not at all related to running beforehand as they are on different planes (i.e.horizontal and vertical) which I thought were independent of each other. I would have thought that if you had no run up at all then the time in the air would be the same you just would have minimal horizontal velocity so you wouldn't jump very far. Also, one other question I had:

Does the run up help at all with the size of the vertical reaction force the runner is able to exert before the jump?

 

[NascentOxygen]

Does the run up help at all with the size of the vertical reaction force the runner is able to exert before the jump?

I think a general answer is that a run up would help him attain a higher altitude if and only if the athlete has some means at "take-off" for converting part of his forward speed into vertical speed. (An aeroplane and a fish can do this.)

 

[Jimmy87]

I think a general answer is that a run up would help him attain a higher altitude if and only if the athlete has some means at "take-off" for converting part of his forward speed into vertical speed. (An aeroplane and a fish can do this.)

How can you convert forward speed into any kind of vertical speed? I thought horizontal and vertical motions are independent of each other so wouldn't that make it impossible for any of the energy associated with the horizontal plane to go into the vertical plane. Surely the vertical push off provides this which has no contributions from the run up? I thought the run up only serves to provide horizontal distance during the hang time?

 

[andrewkirk]

I can't think of any reason why the run up should increase the 'hang time' as they call the time in the air. It doesn't need to, because for a given hang time, the greater the horizontal speed the greater the distance travelled.

It shouldn't be too hard to do some empirical measurement of this. Just get a stopwatch and a video of an International Atheltics competition and compare the average hang times of high-jumpers against long jumpers. To time the high-jumpers you'd need to time from take-off to apex and double it, because they don't fall all the way down (because of the mats). My guess is that the high-jumpers would have longer hang-time, because they are exclusively focused on height. Yet a high-jumper's run-up is much slower than a long-jumper's. If horizontal speed enabled you to go higher, we'd see high-jumpers running up flat-out. But they don't.

A pole vaulter can efficiently convert horizontal momentum into vertical by using the long bendy pole, which can absorb the energy from horizontal motion as it bends, then deliver it as upward motion. An aeroplane can convert horizontal to vertical motion by increasing the angle of attack of its wings, which generates lift that is proportional to velocity (or its square?), at the expense of greater drag. And presumably a fast fish can do the same thing with its fins. But a long-jumper's leg has little-to-no capacity to do the same thing.

 

[HallsofIvy]

The crucial point here is that velocity and position are vector quantities. That is, horizontal and vertical components are independent. The height of the jump depends solely upon the vertical component of force applied at the time of the jump. The length of the jump depends upon the horizontal component of force, if any, and the initial horizontal velocity due to the run up.

 

[Jimmy87]

The crucial point here is that velocity and position are vector quantities. That is, horizontal and vertical components are independent. The height of the jump depends solely upon the vertical component of force applied at the time of the jump. The length of the jump depends upon the horizontal component of force, if any, and the initial horizontal velocity due to the run up.

Perfect, thanks guys. Just one final question. How do you explain the longer time in the air from the larger ground reaction force. I'm not sure which equations/physics to select to show this. For example, I'm getting stuck picking time = distance/speed. A larger ground reaction force would give a larger acceleration and therefore a larger speed. This would give a larger height according to the equation just mentioned. However, wouldn't a larger speed and larger distance result in the same time. I guess these variables are not proportional (i.e. the increase in velocity must be less than the increase in height) since we know the hang time IS longer with a greater ground reaction force. Why doesn't this logic work? If you approach it from an energy point of view then you could say that a larger ground reaction force gives a bigger change in vertical momentum. This larger momentum has to brought to zero and be reversed in direction for the jumper to reach the surface of the earth. Gravity is the force that does this. Since this is a fixed force, it takes longer to do this with a larger momentum. Is that ok?

 

[haruspex]

Perfect, thanks guys. Just one final question. How do you explain the longer time in the air from the larger ground reaction force. I'm not sure which equations/physics to select to show this. For example, I'm getting stuck picking time = distance/speed. A larger ground reaction force would give a larger acceleration and therefore a larger speed. This would give a larger height according to the equation just mentioned. However, wouldn't a larger speed and larger distance result in the same time. I guess these variables are not proportional (i.e. the increase in velocity must be less than the increase in height) since we know the hang time IS longer with a greater ground reaction force. Why doesn't this logic work? If you approach it from an energy point of view then you could say that a larger ground reaction force gives a bigger change in vertical momentum. This larger momentum has to brought to zero and be reversed in direction for the jumper to reach the surface of the earth. Gravity is the force that does this. Since this is a fixed force, it takes longer to do this with a larger momentum. Is that ok?

The other respondents seem to agree that greater running speed cannot increase flight time. I do not know what long jumpers do in practice, but I can offer a mechanism by which it could increase 'hang' time.
Consider a pole vaulter. The pole converts horizontal speed to vertical speed. A long jumper might be able to use his legs partly in the same way, storing the energy in tendons.
That said, it seems wrong to assert that a greater run speed leads to longer in the air without offering some such justification.

 

[andrewkirk]

As usual, wikipedia proved to be a useful resource. From the wiki article on long-jump I learned that long jumpers lower their centre of gravity a couple of steps before take-off, presumably by bending their knees more. That enables them to gain some upward momentum in the last step as they straighten their take-off leg. My guess is that the horizontal velocity detracts from rather than assists this. Someone doing a standing jump does the same thing - crouching down then leaping up. More upward momentum can be achieved from a standing jump because one can get the centre of gravity lower than one can when running. So I'd imagine a standing jump would deliver more hang time.

If a long jumper wore big membranes on her hands that were rolled up but could be unfurled just at the right time they could be used to convert horizontal into vertical motion, as a plane or fish does. But I bet they'd be declared illegal if they aren't already banned. It would be like using flippers in a swimming race.