# Question Video: Calculating the Exhaust Speed Required for a Particular Acceleration of a Variable Mass Object

A rocket of mass 1200 kg is in deep space. The rocket accelerates from a speed of 0.80 km/s to a speed of 1.0 km/s in a time of 5.0 s. Using 50 kg of fuel, what must be the exhaust speed to produce this acceleration?

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### Video Transcript

A rocket of mass 1200 kilograms is in deep space. The rocket accelerates from a speed of 0.80 kilometers per second to a speed of 1.0 kilometers per second in a time of 5.0 seconds. Using 50 kilograms of fuel, what must be the exhaust speed to produce this acceleration?

We can name the exhaust speed we want to solve for đť‘Ł sub đť‘’đť‘Ą. And since weâ€™re told that the rocket is in deep space, that means we can essentially neglect the force of gravity on the rocket. Therefore, the ideal rocket equation applies to describe its motion. This equation tells us that the change in a rocketâ€™s speed is equal to the exhaust velocity of its engines times the natural log of its initial mass over its final mass, including the mass of any lost fuel.

Since we want to solve for the exhaust speed, we rearrange the ideal rocket equation to isolate this term on one side. Regarding the change in the rocketâ€™s velocity Î”đť‘Ł, weâ€™re told the rocket accelerates from 0.80 to 1.0 kilometers per second. In units of meters per second, thatâ€™s a change of 200.

Now we consider the initial and final mass of our rocket, including the fuel. The overall mass of the rocket and fuel after the acceleration happened is given as 1200 kilograms. And since we used 50 kilograms of fuel during the speed-up, that must mean the initial mass was 1200 plus 50, or 1250 kilograms.

Plugging in these values for initial and final mass, we see the units of kilograms cancel out. And weâ€™re ready to calculate the exhaust velocity, đť‘Ł sub đť‘’đť‘Ą. To two significant figures, itâ€™s equal to 4.9 times 10 to the third meters per second, or 4.9 kilometers per second. Thatâ€™s the exhaust speed needed to produce this acceleration.