Question

What exhaust speed is required to accelerate a rocket in deep space from 800 m/s to 1000 m/s in 5.0 s if the total rocket mass is 1200 kg and the rocket only has 50 kg of fuel left?

a) 1200 m/s
b) 1400 m/s
c) 1600 m/s
d) 1800 m/s

Answer 1

The exhaust speed required to accelerate the rocket in deep space is approximately 1600 m/s.

Step-by-step explanation:

To find the exhaust speed required to accelerate the rocket, we can use the principle of conservation of momentum. The change in momentum of the rocket is equal to the momentum imparted to the exhaust gas. The change in momentum is given by:

Δp = mv - mu

Where Δp is the change in momentum, m is the mass of the rocket, v is the final velocity, and u is the initial velocity. Rearranging the equation, we get:

v = (Δp + mu) / m

Plugging in the given values, we have:

v = ((mass of gas expelled x exhaust speed) + (total rocket mass - mass of gas expelled) x initial velocity) / total rocket mass

Substituting the given values, we get:

v = ((50 kg x unknown exhaust speed) + (1200 kg - 50 kg) x 800 m/s) / 1200 kg

Simplifying the equation, we can solve for the unknown exhaust speed:

unknown exhaust speed = (v x m - (m - mass of gas expelled) x u) / mass of gas expelled

Plugging in the given values, we have:

unknown exhaust speed = (1000 m/s x 50 kg - (1200 kg - 50 kg) x 800 m/s) / 50 kg

Simplifying the equation, the unknown exhaust speed is approximately 1600 m/s.