# Velocity of 2 objects of different masses

## Homework Statement

I have 2 boxes A and B and they have a mass of 1kg and 2kg respectively. They are pushed with a force of 10N and 20N horizontally but B has twice the surface area as A. Which object would come to a constant velocity first and which constant velocity would be lower?

F=ma

## The Attempt at a Solution

I'm not sure how to answer this because usually for these questions I'm only given one object to access. Now with 2 objects, i know that B would have twice the resistive force for any specific velocity. So now that I'm pushing them, I'm not sure how the air resistance would change for the 2 boxes.

Because now velocity and surface area both affects the air resistance experienced by the boxes. So for sure B has a greater surface area and both have an increasing velocity. However, I'm not sure whether which box would have a greater or smaller air resistance because the velocity which is affected by the air resistance is affected by velocity and surface area. So now I'm really confused about this whole question. Can someone enlighten me? Thanks :)

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mfb
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I think the question is quite uncelar.
Twice the area on the floor? Do we have resistance there?
Twice the area for air resistance? With the same shape or a different one? Is air resistance relevant at all?

Is there any sketch of the setup?
If not: You can consider the case of a doubled air resistance for box B first, and maybe other setups afterwards as comparison to that case.

I think the question is quite uncelar.
Twice the area on the floor? Do we have resistance there?
Twice the area for air resistance? With the same shape or a different one? Is air resistance relevant at all?

Is there any sketch of the setup?
If not: You can consider the case of a doubled air resistance for box B first, and maybe other setups afterwards as comparison to that case.
Sorry about that. For the set up everything should be the same except the surface area and what I meant was that the surface area of the B (2kg) is twice as large as A where the surface area is of the plane of the box's direction of travel.

So I'm not sure sure which box would reach terminal velocity first. Because B would have twice the resistance for a certain velocity. But now both boxes don't have the same velocity so I'm not sure how to tell which one has more resistance.

mfb
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But now both boxes don't have the same velocity
If air resistance is doubled for B, I don't see why they should have a different velocity.
In general, the box with the smaller final velocity approaches the final velocity quicker. Apart from some weird hypothetical situations, the objects never reach their final velocity exactly, they just come closer and closer.

If air resistance is doubled for B, I don't see why they should have a different velocity.
In general, the box with the smaller final velocity approaches the final velocity quicker. Apart from some weird hypothetical situations, the objects never reach their final velocity exactly, they just come closer and closer.
But isn't air resistance doubled only when the velocity is the same? That's why I'm not so sure how the resistance gets affected with time..

Thanks :)

haruspex
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the surface area of the B (2kg) is twice as large as A where the surface area is of the plane of the box's direction of travel.
You mean, perpendicular to the direction of travel, yes?
If drag is proportional to area then, at the same speed, what's the ratio of the drag forces? What's the ratio of the net forces?

You mean, perpendicular to the direction of travel, yes?
If drag is proportional to area then, at the same speed, what's the ratio of the drag forces? What's the ratio of the net forces?
Thanks for the reply :) and Yup

The ratio of the drag forces would be 2:1. But won't the speed change as well? So if A and B has the same initial acceleration B would get more drag because of the increased surface area. But if we just explain that, won't A have a higher velocity than B?

haruspex
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The ratio of the drag forces would be 2:1.
Yes. And the propulsive forces are 2:1, and the masses are 2:1. So what can we say about the two accelerations?

Yes. And the propulsive forces are 2:1, and the masses are 2:1. So what can we say about the two accelerations?
They should have the same acceleration if that's the only force. But I still can't see how their resistances would be like. Because if I were to say that B has two times the resistance that would mean that A would be faster, but if A was faster that would mean that air resistance should also be greater. But now these processes are simultaneous too. So I'm having trouble understanding that.

Thanks for the hop given :)

mfb
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Because if I were to say that B has two times the resistance that would mean that A would be faster
No. B would just be like two A next to each other: All forces and all masses are doubled.

No. B would just be like two A next to each other: All forces and all masses are doubled.
Oh why would the resistance just be twice in at B when compared to A? Because I thought the velocity would also be a factor in the resistance?

haruspex
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Oh why would the resistance just be twice in at B when compared to A? Because I thought the velocity would also be a factor in the resistance?
There has to be a reason for the velocities to differ first. We have shown that, at the same velocity, the accelerations should be the same, therefore the velocities will always be the same.

There has to be a reason for the velocities to differ first. We have shown that, at the same velocity, the accelerations should be the same, therefore the velocities will always be the same.
Won't the velocities differ because of the difference in resistance? That's why I'm having some trouble understanding this. It's like both velocity and surface area affects the magnitude of resistance, so now that B has more resistance than A it would have a lower velocity than A as it has a smaller acceleration value. But that would also suggest that A is faster and so shouldn't it also mean it has more resistance than B?

I hope that shows the misconception better haha thanks so much

mfb
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Maybe it is easier if we consider an example with numbers:

Let A and B start with the same velocity of 0, and no air resistance (as the velocity is 0). Acceleration of A is 10N/(1kg) = 10m/s^2, acceleration of B is 20N/(2kg) = 10m/s^2.

Later, A and B gained some speed. Let's say A experiences air resistance of 5N. The acceleration of A is (10N-5N)/(1kg) = 5m/s^2. Let's assume that B has the same speed: It experiences air resistance of 10N, and has an acceleration of (20N-10N)/(2kg) = 5m/s^2. As you can see, if the velocity is the same, the acceleration (and therefore the future velocity) will be the same as well. There is no reason why the velocity should differ at any point in time.

haruspex
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Won't the velocities differ because of the difference in resistance?
Think about the chain of causality. Velocities will differ if and only if prior accelerations differed. Accelerations differ only if the ratio of net force to mass differs. If the velocities have been the same up to some point in time t, all forces are in the same ratio as the masses.

Maybe it is easier if we consider an example with numbers:

Let A and B start with the same velocity of 0, and no air resistance (as the velocity is 0). Acceleration of A is 10N/(1kg) = 10m/s^2, acceleration of B is 20N/(2kg) = 10m/s^2.

Later, A and B gained some speed. Let's say A experiences air resistance of 5N. The acceleration of A is (10N-5N)/(1kg) = 5m/s^2. Let's assume that B has the same speed: It experiences air resistance of 10N, and has an acceleration of (20N-10N)/(2kg) = 5m/s^2. As you can see, if the velocity is the same, the acceleration (and therefore the future velocity) will be the same as well. There is no reason why the velocity should differ at any point in time.
Ohh thanks so much for the help. Sorry for the late response needed to integrate into a new school..

What would happen if I had 2 boxes A and B with the same mass of 2kg but B has twice the surface area in the plane of the direction of travel. Because now in this case I'm not too sure how everything would play out..

Thanks so much :)

mfb
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With the same driving force? B would be slower, and approach its (lower) final velocity quicker.

With the same driving force? B would be slower, and approach its (lower) final velocity quicker.
Sorry again for the late reply.

But if B is slower than A won't that also mean that A could have more resistance? That's where i get the problem with this 'paradox'

mfb
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But if B is slower than A won't that also mean that A could have more resistance? That's where i get the problem with this 'paradox'
Sure, but if A is quicker, it will stay quicker forever, with a similar argument as in the original question.