4. A rocket accelerates by burning its onboard fuel, so that its mass decreases with time. Suppose that the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ν (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation v(t) = −ν ln(1 −Rt) −gt, where g is the acceleration due to gravity, R = r/m, and t is not too large. (a) Find an expression for the acceleration, a(t) = v′(t), of the rocket at any time t. (b) Find an expression for the position of the height, h(t), of the rocket at any time t. You can assume that at t = 0, the rocket hasn’t left the ground. Hint: re- member that ν,R,g are all constants and t is the independent variable. Write out your steps carefully! (c) Let g = 9.8 m/s2, R = 0.005 s−1. What does the velocity of the exhaust gases, ν, need to be for the rocket to reach a height of 6000 m one minute after liftoff? You can round to the nearest whole number.

4. A rocket accelerates by burning its onboard fuel, so that its mass decreases with time. Suppose that the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ν (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation v(t) = −ν ln(1 −Rt) −gt, where g is the acceleration due to gravity, R = r/m, and t is not too large. (a) Find an expression for the acceleration, a(t) = v′(t), of the rocket at any time t. (b) Find an expression for the position of the height, h(t), of the rocket at any time t. You can assume that at t = 0, the rocket hasn’t left the ground. Hint: re- member that ν,R,g are all constants and t is the independent variable. Write out your steps carefully! (c) Let g = 9.8 m/s2, R = 0.005 s−1. What does the velocity of the exhaust gases, ν, need to be for the rocket to reach a height of 6000 m one minute after liftoff? You can round to the nearest whole number.