### 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.