# Conservation of energy with kinetic energy and spring

## Homework Statement

A dynamics cart 1 ahs a mass of 1.8 kg and is moving with a velocity of 4.0 m/s
along a frictionless track. Dynamics cart 2 has a mass of 2.2 kg and is moving at 6.0 m/s
. The carts collide in a head-on elastic collision cushioned by a spring with spring constant k = 8.0 x 104 N/m.

Determine the compression of the spring in cm, during the collision when cart 2 is moving at 4.0m/s

and

Calculate the maximum compression of the spring in cm

## The Attempt at a Solution

so the givens are these:
m1 = 1.8 kg
m2 = 2.2 kg
v1=4 m/s

v2 = 6 m/s

K = 80000N/m

a)

Ek = 1/2m2v2
Ek = 1/2(2.2kg)(4m/s
)^2
Ek = 17.6 J

17.6 J = 1/2kx^2
17.6 J = 1/2(80000N/m)x^2
0.020976176 m = x

b)

I am not sure what to do for this particular part?

v1=(m1-m2)/(m1+m2)v1 + (m1xm2)/(m1+m2)v2
v1=0.4 m/s
+5.94 m/s

v1= 6.34 m/s

v2 = (2m1)/(m1+m2)v1 + (m2-m1)/(m1+m2)v2
v2 = 3.6 m/s
+ 0.6 m/s

v2 = 3 m/s

Ektot = 1/2m1v1^2 + 1/2m2v2^2
Ektot = 14.4 J +39.6 J
Ektot = 54 J

Ektot = 1/2m1v1^2 + 1/2m2v2^2
Ektot = 36.17604 J + 9.9 J
Ektot = 46.07604 - but they are not the same since its elastic they are supposed to be the same​

Last edited:

SammyS
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## Homework Statement

A dynamics cart 1 ahs a mass of 1.8 kg and is moving with a velocity of 4.0 m/s
along a frictionless track. Dynamics cart 2 has a mass of 2.2 kg and is moving at 6.0 m/s
. The carts collide in a head-on elastic collision cushioned by a spring with spring constant k = 8.0 x 104 N/m.

Determine the compression of the spring in cm, during the collision when cart 2 is moving at 4.0m/s

and

Calculate the maximum compression of the spring in cm

## The Attempt at a Solution

so the givens are these:
m1 = 1.8 kg
m2 = 2.2 kg
v1=4 m/s

v2 = 6 m/s

K = 80000N/m

a)

Ek = 1/2m2v2
Ek = 1/2(2.2kg)(4m/s
)^2
Ek = 17.6 J

17.6 J = 1/2kx^2
17.6 J = 1/2(80000N/m)x^2
0.020976176 m = x

b)

I am not sure what to do for this particular part?​

For part a), which is incorrect:

What is the KE of cart 1, when cart 2 is moving at 4.0m/s
? You need to include this in your total KE.​

For part a), which is incorrect:

What is the KE of cart 1, when cart 2 is moving at 4.0m/s
? You need to include this in your total KE.​

would it be Ek = 1/2m1v1^2
Ek = 1/2(1.8kg)(4m\s)^2
Ek= 14.4 J

so then itll be

Ektot = 14.4 J + 17.6 J
Ektot = 32 J

32 J = 1/2(80000N/m)x^2
0.02828 m = x

how about this?​

SammyS
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would it be Ek = 1/2m1v1^2
Ek = 1/2(1.8kg)(4m\s)^2
Ek= 14.4 J

so then itll be

Ektot = 14.4 J + 17.6 J
Ektot = 32 J

32 J = 1/2(80000N/m)x^2
0.02828 m = x

how about this?
Incorrect.

Use conservation of momentum to find the velocity of Cart 1, when Cart 2 is moving at 4.0m/s

Incorrect.

Use conservation of momentum to find the velocity of Cart 1, when Cart 2 is moving at 4.0m/s

Okay .. but my other tutor said you can't use momentum because the energy is being conserved into the spring not the balls itself...? But if you suggest it i can try it that way...

It would be m1v1 + m2v2 = m1v1 + m2v2

(1.8 kg)(4 m/s
) + (2.2 kg)(6 m/s
) = (1.8 kg)v1 + (2.2 kg)(4m/s
)
7.2 kgm/s
+ 13.2kgm\s
= (1.8kg)v1 + 8.8kgm/s

6 kgm/s
- 8.8kgm\s
= (1.8kg)v1
2.8 kgm\s
/ 1.8 kg = v1
1.556 m/s
= v1

Ektot= 1/2m1v1^2 + 1/2m2v2^2
Ektot = 2.1790 J + 17.6 J
Ektot = 19.779 J

Ektot = 1/2kx^2
19.779 J / 40000 N/m = x^2
0.022236 m = x​

Incorrect.

Use conservation of momentum to find the velocity of Cart 1, when Cart 2 is moving at 4.0m/s

but it doesn't say that the m1 would also be changing in the velocity.... then why do we need to find the v1 ?​

SammyS
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but it doesn't say that the m1 would also be changing in the velocity.... then why do we need to find the v1 ?
... because the velocity of m1 does change! Conservation of momentum tells you that, without it being stated in the problem .

... because the velocity of m1 does change! Conservation of momentum tells you that, without it being stated in the problem .

Oh i see what you are talking about ..... so how is my part a calculation i just did?

ANY BODY?! SammyS
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ANY BODY?! Patience !

You need to quit prematurely bumping your posts !

SammyS
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Okay .. but my other tutor said you can't use momentum because the energy is being conserved into the spring not the balls itself...? But if you suggest it i can try it that way...

It would be m1v1 + m2v2 = m1v1 + m2v2

(1.8 kg)(4 m/s
) + (2.2 kg)(6 m/s
) = (1.8 kg)v1 + (2.2 kg)(4m/s
)
7.2 kgm/s
+ 13.2kgm\s
= (1.8kg)v1 + 8.8kgm/s

6 kgm/s
- 8.8kgm\s
= (1.8kg)v1
2.8 kgm\s
/ 1.8 kg = v1
1.556 m/s
= v1

Ektot= 1/2m1v1^2 + 1/2m2v2^2
Ektot = 2.1790 J + 17.6 J
Ektot = 19.779 J

Ektot = 1/2kx^2   This is the Potential Energy of the spring.
19.779 J / 40000 N/m = x^2
0.022236 m = x​

Why are you equating the Potential Energy of the spring to the total KE of the two carts?​

Why are you equating the Potential Energy of the spring to the total KE of the two carts?

Well the amount of energy that the carts have would be transferring into the spring wouldn't it?

Why are you equating the Potential Energy of the spring to the total KE of the two carts?

Well when i come to think of it...
Ektot = 19.779 J is the kinetic energy when they have come together compressing the spring..
but how much did they have the initial energy....
To do this i did this:

Ektot(initial) = 1/2m1v1^2 + 1/2m2v2^2
14.4 J + 39.6 J = 54 J

And if i subtract the 54 J - 19.779 J = 34.221 thats the energy lost in the spring correct?

so then i can use that ...

34.221 J = 1/2(80000N/m)x^2
0.029249 m = x

SammyS
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Well the amount of energy that the carts have would be transferring into the spring wouldn't it?
The amount of (kinetic) energy the carts lose will be transferred to PE of the spring. The Total energy is conserved.

The amount of (kinetic) energy the carts lose will be transferred to PE of the spring. The Total energy is conserved.

Yes that is correct similarely what i have done like this..

Ektot(initial) = 1/2m1v1^2 + 1/2m2v2^2
14.4 J + 39.6 J = 54 J

And if i subtract the 54 J - 19.779 J = 34.221 thats the energy lost in the spring correct?

so then i can use that ...

34.221 J = 1/2(80000N/m)x^2
0.029249 m = x

im i correct on this calculation?

SammyS
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Yes that is correct similarely what i have done like this..

Ektot(initial) = 1/2m1v1^2 + 1/2m2v2^2
14.4 J + 39.6 J = 54 J

And if i subtract the 54 J - 19.779 J = 34.221 thats the energy [STRIKE]lost[/STRIKE] in the spring correct?

so then i can use that ...

34.221 J = 1/2(80000N/m)x^2
0.029249 m = x

I'm i correct on this calculation?

Yes. now convert that to cm.

Yes. now convert that to cm.

Okay but now what about the the sexond part where you need to find the maximum spring compression ? i showed my work above... but the energy before and after is different.. is that the way u should do it?

SammyS
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...

Calculate the maximum compression of the spring in cm

## The Attempt at a Solution

so the givens are these:

m1 = 1.8 kg  m2 = 2.2 kg  v1=4 m/s
v2 = 6 m/s
K = 80000N/m

b)

I am not sure what to do for this particular part?

v1=(m1-m2)/(m1+m2)v1 + (m1xm2)/(m1+m2)v2
v1=0.4 m/s
+5.94 m/s

v1= 6.34 m/s

v2 = (2m1)/(m1+m2)v1 + (m2-m1)/(m1+m2)v2
v2 = 3.6 m/s
+ 0.6 m/s

v2 = 3 m/s

Ektot = 1/2m1v1^2 + 1/2m2v2^2
Ektot = 14.4 J +39.6 J
Ektot = 54 J

Ektot = 1/2m1v1^2 + 1/2m2v2^2
Ektot = 36.17604 J + 9.9 J
Ektot = 46.07604 - but they are not the same since its elastic they are supposed to be the same​

Where did you get those equations for v1' & v2'? For what conditions are those equations true? I suspect that they're for the velocities resulting from an elastic collision. As you point out, there's something wrong because there is kinetic energy lost.

While you have an error somewhere when you used these. They have nothing to do with solving part b).

At the instant the spring is at maximum compression, the two carts have the same velocity, v1 = v2 . Why is that the case?​

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Where did you get those equations for v1' & v2'? For what conditions are those equations true? I suspect that they're for the velocities resulting from an elastic collision. As you point out, there's something wrong because there is kinetic energy lost.

While you have an error somewhere when you used these. They have nothing to do with solving part b).

At the instant the spring is at maximum compression, the two carts have the same velocity, v1 = v2 . Why is that the case?

well those equations are true for two moving object.. and i found the mistake where i was wrong.... for the v1 . the second mart is (2m1/m1+m2)v2

i used these to find the kinetic energy i just wanted to see if they equal the kinetic energy before and after.. and its true... so now to find the maximum compression.. i need to find the velocity of them in the middle.... right?

SammyS
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well those equations are true for two moving object.. and i found the mistake where i was wrong.... for the v1 . the second mart is (2m1/(m1+m2))v2

i used these to find the kinetic energy i just wanted to see if they equal the kinetic energy before and after.. and its true... so now to find the maximum compression.. i need to find the velocity of them in the middle.... right?
In the middle of what ?

Use conservation of momentum to find v2 & v2, when they're equal. Of course that's when v1 = v2 = vcenter of mass. And, I asked earlier, if you know why that gives the distance of closest approach.

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In the middle of what ?

Use conservation of momentum to find v2 & v2, when they're equal. Of course that's when v2 v2 = vcenter of mass. And, I asked earlier, if you know why that gives the distance of closest approach.

Alright this is what i did...

m1v1 + m2v2 = (m1+m2) Vmin
(1.8kg)(4m/s
) + (2.2kg)(6m/s
) = ( 4kg ) Vmin
7.2 kgm/s
+13.2kgm/s
/4kg = vmin
1.5 m\s
= Vmin

Ekmin = 1/2m1Vmin^2 + 1/2m1Vmin^2
Ekmin = 1/2(1.8kg)(1.5m\s)^2 + 1/2(2.2kg)(1.5m\s)^2
Ekmin=2.025 J + 2.475 J
Ekmin = 4.5 J

Ektot - Wdef = Ekmin
54 J - Wdef = 4.5 J
Wdef = 49.5 J

Ekspring = 1/2kx^2
Ekspring = 1/2(80000N/m)x^2
49.5 J/40000N/m = x^2
0.035178 m =x
3.178 cm = x​

SammyS
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Why is it that when v1 = v2, then the spring is at maximum compression. In other words, How is it that you know that this is the condition for which the carts are at a minimum distance from each other?

Why is it that when v1 = v2, then the spring is at maximum compression. In other words, How is it that you know that this is the condition for which the carts are at a minimum distance from each other?

Well we did a long prove question in class. and that i found out.. but like in general just think about it .. if the spring is connected to both of them.. then the velocity have to be the same inorder and the distance have to be short inorder to cause the maximum compression of the spring but are my calculation correct u would say?

SammyS
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Well we did a long prove question in class. and that i found out.. but like in general just think about it .. if the spring is connected to both of them.. then the velocity have to be the same inorder and the distance have to be short inorder to cause the maximum compression of the spring but are my calculation correct u would say?
They look fine to me.