Simple p=mv question needs simple answer

[Tranesblues]

I realize that this is a very simple question but I need a dispute resolved. Please provide the simplest answer you can to the following question. I realize that friction and subtle things come into play in a question like this but please avoid it in your response. I am looking for the simplest answer possible. I am an 8th grade teacher and I need an answer like you would expect from and 8th grader or an 8th grade teacher. Thanks.

If a truck and a bicycle have the same momentum, which one is harder to stop?

 

[Nugatory]

I
If a truck and a bicycle have the same momentum, which one is harder to stop?

Assuming that the truck is much heavier than the bicycle...

Any non-zero force will eventually stop both the truck and the bicycle. If you apply the same force to slowing both, the truck will travel much farther and it will take much longer before it stops.

 

[Nugatory]

If they have the same momentum, doesn't that mean that neither is harder to stop? In this case we are told nothing else.

Sorry, you're right. I answered for "same initial speed".

 

[Tranesblues]

But if two objects have the same momentum, arent they equally hard to stop? In this case we are told nothing about mass or velocity.

 

[Q_Goest]

For a truck and a bike to have the same momentum, the mass of the vehicle times the velocity of the vehicle must be the same for both of them. Therefore, the lighter, less massive bike will have a much higher velocity than the truck.

The phrase "harder to stop" however, has to be interpreted somehow. Energy is force times distance, so I think one reasonable interpretation of the phrase would be to suggest that given you try to stop these two vehicles over the same distance, then whichever requires the greater force would be "harder to stop". In other words, you would have to push harder (exert more force) on the one that's harder to stop in order to stop it in the same distance. To determine that, we need to know how much energy each of them has at the given velocity. In this case, we're looking at kinetic energy.

The kinetic energy is a function of the mass times the velocity squared, or to put it another way, it is a function of the momentum times the velocity. So if the momentum is the same for both but the velocity is much higher for the bike than the truck, then the kinetic energy is greater for the bike. So if we use the concept that the one that's harder to stop has more energy, then the bike is harder to stop because we have to take more energy out of the bike to stop it, and if we take that energy out over a given distance for each of these, the force we need to stop the bike is greater.

It could be that other concepts of what is meant by "harder to stop" might yield a different answer.

 

[Tranesblues]

Sorry, you're right. I answered for "same initial speed".

Sorry, I think i went to edit my post and we got crossed up. Thanks for the quick reply.

 

[Tranesblues]

For a truck and a bike to have the same momentum, the mass of the vehicle times the velocity of the vehicle must be the same for both of them. Therefore, the lighter, less massive bike will have a much higher velocity than the truck.

The phrase "harder to stop" however, has to be interpreted somehow. Energy is force times distance, so I think one reasonable interpretation of the phrase would be to suggest that given you try to stop these two vehicles over the same distance, then whichever requires the greater force would be "harder to stop". In other words, you would have to push harder (exert more force) on the one that's harder to stop in order to stop it in the same distance. To determine that, we need to know how much energy each of them has at the given velocity. In this case, we're looking at kinetic energy.

The kinetic energy is a function of the mass times the velocity squared, or to put it another way, it is a function of the momentum times the velocity. So if the momentum is the same for both but the velocity is much higher for the bike than the truck, then the kinetic energy is greater for the bike. So if we use the concept that the one that's harder to stop has more energy, then the bike is harder to stop because we have to take more energy out of the bike to stop it, and if we take that energy out over a given distance for each of these, the force we need to stop the bike is greater.

It could be that other concepts of what is meant by "harder to stop" might yield a different answer.

Awesome. Thanks. What answer would you expect from and 8th grader on this? I have put it there as a bonus question.

 

[technician]

To stop requires the same CHANGE in momentum for each object.
Stopping requires a FORCE to act on each object
CHANGE in MOMENTUM = Force x TIME for which the force acts
So the answer is not simple !
A large force acting for a small time will cause the same effect as a small force acting for a long time.
This sounds like a question used to start a discussion.
Hope this helps

 

[Q_Goest]

Awesome. Thanks. What answer would you expect from and 8th grader on this? I have put it there as a bonus question.

I could see an 8'th grader suggesting that the amount of force needed to stop a truck will be larger than that needed to stop a bike, without giving it too much thought. In that case, the intuition is that there's a force needed to slow the vehicle at a given rate. So the amount of force needed to decelerate a truck at a given rate would obviously be much greater than the force needed to decelerate a bike, so they might say it is harder to stop the truck.

However you respond to this question, if someone were to define how they interpreted the phrase "harder to stop" and gave the right answer given their interpretation, I'd say that person should get the bonus.

 

[Tranesblues]

However you respond to this question, if someone were to define how they interpreted the phrase "harder to stop" and gave the right answer given their interpretation, I'd say that person should get the bonus.

I am expecting that most will say the truck b/c it has more mass. I am not sure that is correct. But I am certainly not a physicist. The answer in my mind, again, on an 8th grade level, is 'neither.' If two objects have the same momentum, is that not the same as saying it takes the same force to stop them? We know nothing else in this case. I realize the question, at its core, is much more complicated than this, but I am hoping to nail down a correct answer between 'the truck,' 'the bike' and 'neither.' Thanks again for the help.

 

[jbriggs444]

Momentum scales directly with speed. Energy scales with the square of speed. If the momentum of the truck and the bicycle are equal then the bicycle must be moving faster than the truck. It must also have more energy than the truck.

In order to stop the bicycle you will need to absorb more energy. Whether that's "easier" to do or "harder" depends on how you go about it.

If you are comparing a 10,000 kilogram truck with a 100 kilogram cyclist then the cyclist will be going 100 times faster than the trucker and will have 100 times more kinetic energy. If the trucker is approaching a brick wall at 1 meter per second then the cyclist must be approaching the wall at 100 meters per second (over 200 miles per hour).

If the truck starts 100 meters away from the wall, that gives you at least 100 seconds to slow it to a stop before it hits (200 seconds if it decellerates evenly for an average speed of 0.5 meters/second). A momentum of 10,000 kilogram meters/sec divided by 200 seconds is 50 Newtons of force. That's about 5 pounds (force).

If the cyclist starts 100 meters away from the same wall, that gives you one second (two seconds if it decellerates evenly for an average speed of 50 meters/second). A momentum of 10,000 kilogram meters/sec divided by 2 seconds is 5000 Newtons of force. That's about 500 pounds (force).

Which do you think would be easier to apply?

 

[Q_Goest]

If two objects have the same momentum, is that not the same as saying it takes the same force to stop them?

Saying they have the same momentum only means that if you multiply the mass times velocity of the two, they will be the same. The mass of the truck times its velocity would be equal to the mass of the bike times its velocity.

What you mean by "harder to stop" makes all the difference in the world. How would you interpret that phrase?

 

[technician]

If you exert the same force on each they will take the same time to come to rest but they will travel different distances.
You could say they are equally easy to stop! (same time)
Or one is harder to stop (one travels further)

Good question...isn't it?
Will raise more questions than it answers !

 

[Tranesblues]

If you exert the same force on each they will take the same time to come to rest but they will travel different distances.
You could say they are equally easy to stop! (same time)
Or one is harder to stop (one travels further)

I think this answer gets down to my ideas on it. I am really probably going to give credit for this answer to anyone who answers it at all. The thing I want them to understand is that just b/c the truck is larger doesn't automatically mean it requires more force. I see now that the term 'harder' is problematic and will change it in the future. In my mind, I read it in the way you have answered it here. Meaning, if you apply the same force (meaning if you, personally, were to try and stop these objects) then it would take the same amount of time to stop. the distance may be different but equal force would stop them at the same time. thanks again for all the input. I am hoping that this will spark a conversation. It's tough getting 8th graders to talk about physics, but anything that challenges their assumptions usally gets them going.

 

[technician]

That's great !
It is a very good question and would cause lively debate at any level, I would love to be a fly on the wall when your kids are discussing this. Have fun

 

[BobG]

Force measures your change in momentum, which means:

F = \Delta p =\Delta (mv)

Assuming your mass doesn't change (which is the usual assumption), that means:

F = m (\Delta v) = ma

To have the same momentum, the bicycle must have a higher velocity. If the acceleration (or deceleration in this case, since the acceleration is negative) is the same for both, which is going to take the longer time (and distance) to stop?

This is also consistent with the difference in energy between the two objects, since energy is equivalent to work, which is Force times distance.

Either you need more force to stop the bicycle in the same amount of time/distance or you need more time/distance to stop the bicycle with the same amount of force.

Either way, I'd say the bicycle is harder to stop.

Part of the difficulty in visualizing this could be easier to see if you used actual numbers. If the truck were 5000 kg and traveling about 20 m/s (about 40 mph), then your momentum would be 100,000 kg-m/s. To have the same momentum, a 10 kg bicycle would have to be traveling 10,000 m/s (about 20,000 mph). Escape velocity from Earth orbit is only a little over 11,000 m/s. Most people might have a problem visualizing that sort of speed. For any situation the average person could visualize, common sense correctly tells them the truck will have much more momentum and be much harder to stop.

I guess the problem proves a point, but the example is so absurd that it's practically guaranteed to encourage wrong answers.

Unless you describe the situation more accurately. If you try to travel 10,000 m/s through the Earth's atmosphere, people probably won't see a bicycle coming at them. They'll see a glowing, flaming fireball coming at them at an incredibly high rate of speed (they won't hear it coming since it's traveling many times the speed of sound). They might be a little more likely to deduce it's going to be even harder to stop that flaming fireball than to stop a truck.

 

[technician]

I guess the problem proves a point, but the example is so absurd that it's practically guaranteed to encourage wrong answers.

It will encourage answers and debate. Whether the answers are right or wrong will come out in the debate.

5000kg truck going at 1m/s can be visualised...big truck going dead slow.
10kg bike going at 500m/s can be visualised ...little bike going very fast

Kids have no problem thinking this way.
They will love discussing it. With encouragement they might even become interested in physics.
Be grateful for absurd examples.

PS ...and innovative teachers, prepared to take a risk at getting a wrong answer

 

[BobG]

And just to help visualize, a 5000 kg truck is a big pick-up truck, such as the Dodge Ram 3500, a Ford F-350, or GMC Sierra 3500, or a Hummer H1.

For a loaded semi-trailer truck, the mass could get up to 40,000 kg without a special permit (too heavy and the road wear and damage gets too great). But 80,000 m/s carries enough kinetic energy to escape the solar system, let alone Earth orbit.

 

[technician]

For any situation the average person could visualize, common sense correctly tells them the truck will have much more momentum and be much harder to stop.

This question specifically refers to 2 objects with the same momentum. It is not about common sense, it is about physics principles. I, and probably the kids, have seen contestants on Iron man type programmes pulling massive trucks at about 0.5m/s

They'll see a glowing, flaming fireball coming at them at an incredibly high rate of speed
What do you mean by 'high rate of speed' ??

A transit van has a mass of about 5000kg

 

[BobG]

They'll see a glowing, flaming fireball coming at them at an incredibly high rate of speed
What do you mean by 'high rate of speed' ??

In a very flat area, the horizon is close to 5 km away. It would take less than half a second for the bicycle (flaming fireball) to go from the horizon to you. You wouldn't have time to involuntarily soil yourself.

In fact, at that velocity, if the bicycle hit you, the collision would be over before your toes and fingers even knew there was a collision. You'd shatter instead of be torn apart (or at least the part of your body involved in the collision would shatter). The bicycle would also shatter, but its shattered pieces would still keep on going in slightly different directions at a slightly slower velocity. (This also happens in collisions between satellites lest you think I'm just the gruesome sort that's into studying shattering body parts.)

 

[technician]

Still not certain what you mean by 'high rate of speed'...I do not think that it is a recognised physics term...do you mean 'acceleration'?
This is a physics forum and you would expect correct terminology to be used.
A bicycle traveling at 500 m/s would not produce the fictional scenario you describe.(probably would not be heard but it would be observed quite easily.
Anyway, you have gone well away from the physics principles raised in this post.
Please stick to sensible physics contributions within the context of the original post.

 

[BobG]

Still not certain what you mean by 'high rate of speed'...I do not think that it is a recognised physics term...do you mean 'acceleration'?
This is a physics forum and you would expect correct terminology to be used.

Incredibly high rate of speed, as in a velocity vector with an incredibly large magnitude that stays constant with no change in direction. In precise terms (or not so precise terms), 20 times faster than a B-58 bomber.

I would have said acceleration (or perhaps even a high rate of acceleration or low rate of acceleration) if I meant the speed changed.

Admittedly, hypervelocity would be a better term than high rate of speed, as high rate of speed is a bit of an understatement for something like 10,000 m/s. A paltry 500 m/s (the speed of a B-58 bomber) is a high rate of speed.

But keep in mind that the term "high rate of speed" was used a mere single sentence away from the actual number I was talking about (with a "mere single sentence" being a much more precise measurement than a plain old "single sentence" away, although I might have trouble locating the precise conversion factor between the two). I doubt my wording confused very many.

If you try to travel 10,000 m/s through the Earth's atmosphere, people probably won't see a bicycle coming at them. They'll see a glowing, flaming fireball coming at them at an incredibly high rate of speed ...

But the example is starting to trample the original question and answer. While "harder to stop" may be an imprecise way to ask the question, I think the answer is obviously the bicycle for most normal interpretations of "harder to stop".

 

[BobG]

In any event, the aim of these problems should be to show how physics explains the world around them; not to show how physics gets you wacky results, so the examples should be somewhat straight forward. Not necessarily obvious, but it shouldn't take too much to see why an unexpected result wasn't so unexpected after all (which is why I'd prefer the flaming fireball of a bicycle to a supersonic bicycle, even though I don't generally like the idea of telling students that a fast moving bicycle is harder to stop than a slow moving truck when they'll never see any situations where that's actually true).

If I were covering this problem I'd start with some actual mass, velocity, and momentum (whatever I chose those to be) using p = mv.

Then I'd show how some force would affect them (F=ma). The larger, slower moving object would have a less negative acceleration than the smaller, faster moving object.

After that, it gets harder depending just what you're covering for an 8th grade class. Using v_f = v_i + at, you could figure out how long it takes to bring each object to a stop (it would be the same time).

Using s_f = s_i + vt + 1/2at^2, you could figure out what distance it took to stop each object (it takes a longer distance to stop the smaller, faster object).

The work done (the energy it took to stop each) would be the force times the distance (W=Fd), with the force being applied over a longer distance for the smaller object.

Then I'd calculate the initial kinetic energy of each object using KE=1/2 mv^2, showing how that calculation is exactly the same as the force times distance.

I'm not positive all six of those equations are explained in 8th grade, which makes it even more important to choose an example they can actually visualize.

 

[Redbelly98]

Saying they have the same momentum only means that if you multiply the mass times velocity of the two, they will be the same. The mass of the truck times its velocity would be equal to the mass of the bike times its velocity.

What you mean by "harder to stop" makes all the difference in the world. How would you interpret that phrase?

When I read the OP's post, that was the question I thought of as well.

If you exert the same force on each they will take the same time to come to rest but they will travel different distances.
You could say they are equally easy to stop! (same time)
Or one is harder to stop (one travels further)

Exactly. That sums up the "answer" in a nutshell. For those who like to see that math worked out...

Since the impulse gives the change in momentum:
F \ \Delta t = \Delta p\Delta t = \frac{\Delta p}{F}
So Δt is the same for both, assuming the same force for both cases.

Since the work done gives the change in kinetic energy:
F \ \Delta x = \Delta K = \frac{\Delta(p^2)}{2m}\Delta x = \frac{\Delta(p^2)}{2mF}
So Δx is greater for a smaller mass.

Not sure if or how this could be related at an 8th grade level, since this is usually taught to 10th-12th graders. They would need to understand impulse, work, and kinetic energy in addition to forces and momentum.

Force measures your change in momentum, which means:

F = \Delta p =\Delta (mv)

But it is really impulse that gives the change in momentum. Force gives the rate of change of momentum. So that should be F \Delta t = \Delta p =\Delta (mv)

Assuming your mass doesn't change (which is the usual assumption), that means:

F = m (\Delta v) = ma

To have the same momentum, the bicycle must have a higher velocity. If the acceleration (or deceleration in this case, since the acceleration is negative) is the same for both, which is going to take the longer time (and distance) to stop?

Why do you say the acceleration is the same both? That's certainly not true if the same amount of force is applied to objects of different mass.

This is also consistent with the difference in energy between the two objects, since energy is equivalent to work, which is Force times distance.

Either you need more force to stop the bicycle in the same amount of time/distance or you need more time/distance to stop the bicycle with the same amount of force.

Not quite. They would stop in the same amount of time if the same force is applied, as technician said earlier. See my simple derivation above -- the same momentum and force mean the same stopping time.