- #1

ruairilamb

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- Homework Statement
- A rocket of initial mass, ##m_{i}##, starts at rest and propels itself by an engine ejecting photons backwards. After the rocket has reached a speed of v, it switches off its engine and its final mass is ##m_{f}## . By using four-momenta and three-momenta for the rocket and photon beam, calculate the ratio of the initial to final mass for the rocket as a function of β only.

- Relevant Equations
- ##β = \frac{v}{c}##

conservation of four momentum

conservation of 3 momentum

conservation of energy

Before the engine is switched off: $$P_{Initial rocket} = (m_{i}c, 0)$$

$$P_{photons} = (E/c, -p_{photons})$$

where E is the energy of the photons and ##p_{photons}## is the 3 momentum of the photons.

Rocket after engine switched off: $$P_{Final rocket} = (m_{f}c, m_{f}v)$$

By conservation of four momentum:

$$P_{Initial rocket} + P_{photons} = P_{Final rocket}$$

So, ##(m_{i}c, 0) + (E/c, -p_{photons}) = (m_{f}c, m_{f}v)##

Now, ##E = p_{photons}c## so,

$$(m_{i}c, 0) + (p_{photons}, -p_{photons}) = (m_{f}c,m_{f}v)$$

##p_{photons}## can be written as ##p_{photons} = m_{i}β##

Hence, $$(m_{i}c, 0) + (m_{i}β, -m_{i}β) = (m_{f}c, m_{f}v)$$

I am not sure how to proceed from here, or even if my solution so far is remotely right. If I equate the components of each of the terms I get:

$$m_{i}c + m_{i}β = m_{f}c$$

This implies a ratio of $$\frac{m_{i}} {m_{f}} = \frac{1} {1+β/c}$$

$$P_{photons} = (E/c, -p_{photons})$$

where E is the energy of the photons and ##p_{photons}## is the 3 momentum of the photons.

Rocket after engine switched off: $$P_{Final rocket} = (m_{f}c, m_{f}v)$$

By conservation of four momentum:

$$P_{Initial rocket} + P_{photons} = P_{Final rocket}$$

So, ##(m_{i}c, 0) + (E/c, -p_{photons}) = (m_{f}c, m_{f}v)##

Now, ##E = p_{photons}c## so,

$$(m_{i}c, 0) + (p_{photons}, -p_{photons}) = (m_{f}c,m_{f}v)$$

##p_{photons}## can be written as ##p_{photons} = m_{i}β##

Hence, $$(m_{i}c, 0) + (m_{i}β, -m_{i}β) = (m_{f}c, m_{f}v)$$

I am not sure how to proceed from here, or even if my solution so far is remotely right. If I equate the components of each of the terms I get:

$$m_{i}c + m_{i}β = m_{f}c$$

This implies a ratio of $$\frac{m_{i}} {m_{f}} = \frac{1} {1+β/c}$$

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