# How to find the energy of an object that was at rest?

#### Davidllerenav

1. The problem statement, all variables and given/known data
In two rockets, one of which moves and the other is at rest, the motors are connected for a short time. During their operation they throw the same mass of gas (small in comparison with the mass of the rocket) at the same speed with respect to the rockets. The kinetic energy of the rocket in motion was $E_0$, increase by 4 percent. Determine the energy of the second rocket

2. Relevant equations
$E_k=\frac{1}{2}mv^2$

3. The attempt at a solution
I first wrote the final kinetic energy of the second rocket: $E_k=E_0+\frac{4}{100}E_0$, then by using that equation: $\frac{1}{2}mv_f^2=\frac{1}{2}mv_0^2+\frac{1}{25}(\frac{1}{2}mv_0^2)=mv_f^2=mv_0^2+\frac{1}{25}mv_0^2\Rightarrow m_vf^2=\frac{25mv_0^2+mv_0^2}{25}=\frac{26mv_0^2}{25}\Rightarrow 25v_f^2=26v_0^2$. After that I don't know what to do.

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#### haruspex

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1. The problem statement, all variables and given/known data
In two rockets, one of which moves and the other is at rest, the motors are connected for a short time. During their operation they throw the same mass of gas (small in comparison with the mass of the rocket) at the same speed with respect to the rockets. The kinetic energy of the rocket in motion was $E_0$, increase by 4 percent. Determine the energy of the second rocket

2. Relevant equations
$E_k=\frac{1}{2}mv^2$

3. The attempt at a solution
I first wrote the final kinetic energy of the second rocket: $E_k=E_0+\frac{4}{100}E_0$, then by using that equation: $\frac{1}{2}mv_f^2=\frac{1}{2}mv_0^2+\frac{1}{25}(\frac{1}{2}mv_0^2)=mv_f^2=mv_0^2+\frac{1}{25}mv_0^2\Rightarrow m_vf^2=\frac{25mv_0^2+mv_0^2}{25}=\frac{26mv_0^2}{25}\Rightarrow 25v_f^2=26v_0^2$. After that I don't know what to do.
What relates the change in speed of the rocket to the mass and relative speed of the exhaust?

#### Davidllerenav

What relates the change in speed of the rocket to the mass and relative speed of the exhaust?
I don't know.

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#### Davidllerenav

A conservation law perhaps?
Oh, the conservation of the energy?

#### haruspex

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Oh, the conservation of the energy?
No, there is unknowable energy input from the combustion of the fuel. What does that leave?

#### Davidllerenav

No, there is unknowable energy input from the combustion of the fuel. What does that leave?
the
Potential energy?

#### haruspex

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the
Potential energy?
No, not energy. What other conservation laws do you know?

#### Davidllerenav

No, not energy. What other conservation laws do you know?
Momentum conservation.

#### haruspex

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Momentum conservation.
Right.
You are told
they throw the same mass of gas ...at the same speed with respect to the rockets.
What does that tell you about the changes in momentum?

#### Davidllerenav

What does that tell you about the changes in momentum?
That momentum increases?

#### haruspex

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That momentum increases?
Well, yes, ejectng the exhaust gases increases the momentum of the rocket - that's how rockets work.
But by how much is the momentum of the rocket increased if a mass M of gas is exhausted at speed v relative to the rocket?

#### Davidllerenav

Well, yes, ejectng the exhaust gases increases the momentum of the rocket - that's how rockets work.
But by how much is the momentum of the rocket increased if a mass M of gas is exhausted at speed v relative to the rocket?
It is increasing by 4%, so it would be $P_f=P_i+\frac{1}{25}P_i$, right?

#### jbriggs444

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The kinetic energy of the moving rocket has increased by 4%. That does not mean that its momentum has increased by 4%.

$25v_f^2=26v_0^2$
That should allow you to compute by how much the velocity of the moving rocket had increased.

#### Davidllerenav

The kinetic energy of the moving rocket has increased by 4%. That does not mean that its momentum has increased by 4%.

That should allow you to compute by how much the velocity of the moving rocket had increased.
From that I would have $v_f=\sqrt{\frac{26}{25}v_0\Rightarrow v_f=1.01v_0$, am I right?

#### haruspex

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From that I would have $v_f=\sqrt{\frac{26}{25}}v_0\Rightarrow v_f=1.01v_0$, am I right?
Yes, except that 1.01 is rather inaccurate. Anyway, don't bother about the arithmetic for now. What does momentum conservation allow you to deduce?

#### Davidllerenav

Yes, except that 1.01 is rather inaccurate. Anyway, don't bother about the arithmetic for now. What does momentum conservation allow you to deduce?
Well, since the momentum of the system is constant, so the sum of the both rockets momentum must be constant, right?

#### haruspex

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Well, since the momentum of the system is constant, so the sum of the both rockets momentum must be constant, right?
The two rockets are in different systems. The untethered rocket, together with its exhaust gases, is an isolated system.

#### Davidllerenav

The two rockets are in different systems. The untethered rocket, together with its exhaust gases, is an isolated system.
Then the momentum of the gas should be the same as the momentum increase.

#### haruspex

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Then the momentum of the gas should be the same as the momentum increase.
Right, and what do we know about the momentum of the gas from the other rocket?

#### Davidllerenav

Right, and what do we know about the momentum of the gas from the other rocket?
Well, since the rocket was at rest, the momentum of the system must be zero, so after the exhaust, the momentum of the gas must be the same as the momentum of.tje rocket but negative.

#### haruspex

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Well, since the rocket was at rest, the momentum of the system must be zero, so after the exhaust, the momentum of the gas must be the same as the momentum of.tje rocket but negative.
Yes, but can you connect this with the momentum change of the first rocket?

#### Davidllerenav

Yes, but can you connect this with the momentum change of the first rocket?
Yes. Since both rockets connect their engines at the same time and thy exhaust the same mass of gas, then the change of momentum of the first rocket must be the same as the change of the rocket at rest, right?

#### haruspex

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Yes. Since both rockets connect their engines at the same time and thy exhaust the same mass of gas, then the change of momentum of the first rocket must be the same as the change of the rocket at rest, right?
Yes.

#### Davidllerenav

So, to calculate the momentum of the rocket at rest I just need to use the change in the velocity of the second rocket, which was $1.01v_i$? $P=mv_f=m(1.01v_i)$, and so the kinetic energy from the rocket at rest would be $E_k=\frac{1}{2}m(1.01v_i)^2$. But why did you said that it is rather inaccurate?

"How to find the energy of an object that was at rest?"

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