Question

A 2500 kg rocket is released from space station. As it burns fuel, the rocket's mass decreases and its velocity increases. Let v(m) be the velocity (in meters per second) as a function of mass m. Find the velocity when m = 1444 kg if dv/dm = -40m^(-1/2). Assume that v(2500) = 0.

Answer 1

The velocity of a 2500 kg rocket given dv/dm and the condition that v(2500 kg) = 0 is determined by integrating dv/dm from the start mass to 1444 kg and using the given conditions to find the integration constant.

Step-by-step explanation:

The student is asking about finding the velocity of a rocket given its mass and the derivative of velocity with respect to mass. Given that dv/dm = -40m-1/2 and the initial conditions v(2500 kg) = 0, we can integrate the expression to find the velocity function v(m). As v(m) is the integral of dv/dm with respect to m, we obtain:

v(m) = ∫ dv/dm dm = ∫ (-40m-1/2) dm

Performing the integration between the limits m = 2500 kg and m = 1444 kg, we calculate the change in velocity. The integration results in:

v(m) = -80m1/2 + C

To find the constant C, we use the initial condition v(2500 kg) = 0:

0 = -80(2500)1/2 + C

C = 80(2500)1/2

Substituting back into the equation for velocity v(m) and evaluating at m = 1444 kg:

v(1444 kg) = -80(1444)1/2 + 80(2500)1/2

Finally, computing the numerical value gives the velocity of the rocket when its mass is 1444 kg.