# Two carts are forced apart by a compressed spring

• paulimerci

#### paulimerci

Homework Statement
Two carts are held together. Cart 1 is more massive than Cart 2. As they are forced apart by a compressed spring between them, which of the following will have the same magnitude for both carts.
(A) change of velocity (B) force (C) speed (D) velocity
Relevant Equations
conservation of momentum and Impulse
It's an explosion problem.
When two carts are pulled apart, the bigger one takes longer than the smaller one. So the velocity of the bigger one is small, and the velocity of the smaller one is large, and they are opposite each other. So the momentum before the explosion must be equal to the momentum after the explosion.
The energy is conserved if no energy is lost due to friction or any other external force. If the pulling force is an external force applied to the system, is energy conserved?
The force applied to both the carts should be different if the time taken for the carts to be pulled apart is different. And so the impulse is different for both carts.
I don't know how to interpret the following options given: I know some of the interpretations were wrong, and I would greatly appreciate it if anyone could explain in detail.

Your problem statement says the carts are PUSHED apart by a compressed spring but your discussion says that they are PULLED apart, and apparently by different forces. Do you see the contradiction?

• paulimerci
And there was an explosion! Yay!

...
The energy is conserved if no energy is lost due to friction or any other external force. If the pulling force is an external force applied to the system, is energy conserved?
You can consider the energy-conservative system as cart 1 - spring - cart 2.
In that way, the elastic energy contained in the compressed spring remains within the system all the time.
The force applied to both the carts should be different if the time taken for the carts to be pulled apart is different. And so the impulse is different for both carts.
Reconsider this.
You can't push a heavy piece of furniture is you are standing on a slippery surface.
In order to effectively push in one direction, the spring must be supported or pushed at its other end by a reactive force.

• paulimerci
Your problem statement says the carts are PUSHED apart by a compressed spring but your discussion says that they are PULLED apart, and apparently by different forces. Do you see the contradiction?
Oh I read the question wrong. If it pushed by the spring then it should have exerted an equal and opposite force on the cart? Right?

Oh I read the question wrong. If it pushed by the spring then it should have exerted an equal and opposite force on the cart? Right?
Right.

• phinds
You can consider the energy-conservative system as cart 1 - spring - cart 2.
In that way, the elastic energy contained in the compressed spring remains within the system all the time.

Reconsider this.
You can't push a heavy piece of furniture is you are standing on a slippery surface.
In order to effectively push in one direction, the spring must be supported or pushed at its other end by a reactive force.
Thank you. You mean stored EPE is transferred to a larger mass with less energy and a smaller mass with more energy, and the system's energy is conserved?
Since the objects have varying masses, they will move with different velocities, even though the carts experience the same force, so the options for c and d are wrong, right?

Last edited:
Consider two carts with masses m and 2m and their respective velocities as ##v## and ##\frac{v
}{2}##.
$$K.E_m = \frac{1}{2}mv^2$$
$$K.E_2m = \frac{mv^2}{4}$$
The two carts doesn’t acquire equal K.E's. right?

Thank you. You mean stored EPE is transferred to a larger mass with less energy and a smaller mass with more energy, and the system's energy is conserved?
Since the objects have varying masses, they will move with different velocities, even though the carts experience the same force, so the options for c and d are wrong, right?
If you are pushing two heavy objects simultaneously in opposite directions, how could you transfer more energy to one than to the other?

Use Newton's third law for action and reaction forces.
Use the second law for comparing the acceleration gained by each cart during the time the common spring is pushing on them.

The cart accelerating the most will be the first losing contact with the spring.
At that instant, the spring will lose its support to push the second cart any longer.
Therefore, the transfer of mechanical energy will beigin and end simultaneously for both carts.

Consider two carts with masses m and 2m and their respective velocities as ##v## and ##\frac{v
}{2}##.
$$K.E_m = \frac{1}{2}mv^2$$
$$K.E_2m = \frac{mv^2}{4}$$
The two carts doesn’t acquire equal K.E's. right?

If you are pushing two heavy objects simultaneously in opposite directions, how could you transfer more energy to one than to the other?
I hope you are not suggesting it cannot be done.

• Lnewqban
If you are pushing two heavy objects simultaneously in opposite directions, how could you transfer more energy to one than to the other?

Use Newton's third law for action and reaction forces.
Use the second law for comparing the acceleration gained by each cart during the time the common spring is pushing on them.

The cart accelerating the most will be the first losing contact with the spring.
At that instant, the spring will lose its support to push the second cart any longer.
Therefore, the transfer of mechanical energy will beigin and end simultaneously for both carts.
Your argument is not convincing .

The magnitude of the forces are identical so the lighter cart experiences more acceleration. This means it travels a larger distance. Since the magnitude of the forces are identical, this means more work is performed on it.

• Lnewqban, jbriggs444, SammyS and 1 other person
I see the answer is "B" for post #1.

I'm trying to solve mit problem that is similar to this. which you'll find below. I should have posted this in a separate homework statement, but since this looks similar, I thought I could try it here. My apologies!

Two toy cars with different masses originally at rest are pushed apart by a spring between them. Which TWO of the following statements be true? (A) both toy cars will acquire equal but opposite momenta (B) both toy cars will acquire equal kinetic energies (C) the more massive toy car will acquire the least speed (D) the smaller toy car will experience an acceleration of the greatest magnitude.
And the answer was found to be B, which contradicts post 8. I see that all the answers look correct except for B.

• Lnewqban
I see the answer is "B" for post #1.

I'm trying to solve mit problem that is similar to this. which you'll find below. I should have posted this in a separate homework statement, but since this looks similar, I thought I could try it here. My apologies!

Two toy cars with different masses originally at rest are pushed apart by a spring between them. Which TWO of the following statements be true? (A) both toy cars will acquire equal but opposite momenta (B) both toy cars will acquire equal kinetic energies (C) the more massive toy car will acquire the least speed (D) the smaller toy car will experience an acceleration of the greatest magnitude.
And the answer was found to be B, which contradicts post 8. I see that all the answers look correct except for B.
I agree with you. But if it asks which two, how come the answer it gives is B only?

• nasu
Okay,
https://web.mit.edu/~yczeng/Public/WORKBOOK 1 FULL.pdf
The question is on page 179 (Question No. 15), and the answer is on page 206.
The text that goes with the answer clearly states that the momenta are equal and opposite but the energies differ. So I would guess the question was supposed to ask which one is false.

• paulimerci
Thank you everyone. I truly appreciate!

• Lnewqban
The text that goes with the answer clearly states that the momenta are equal and opposite but the energies differ. So I would guess the question was supposed to ask which one is false.
Adding the kinetic energies of the two masses is equal to the EPE of the spring initial and thus energy is conserved in the system. Right?

Adding the kinetic energies of the two masses is equal to the EPE of the spring initial and thus energy is conserved in the system. Right?
Yes, but how is that relevant?

Yes, but how is that relevant?
It's not relevant. I'm just clarifying myself.

Adding the kinetic energies of the two masses is equal to the EPE of the spring initial and thus energy is conserved in the system. Right?
Correct!

• paulimerci