QP2

As shown in the sketch above, a spaceship on a distant planet has landed on a ledge above a vast flat horizontal plain, a distance L = 100 km from a volcano. Just as the astronaut team emerges to survey the area, a boulder is ejected at t = 0 from the cone of the volcano, at the same height as the ledge, with an initial speed v₀ = 1 km/sec and an intial angle ₀ with the horizontal. Measurements by the team show that the boulder is following a parabolic trajectory, that it reaches its maximum height at t = 200 sec, and that it is headed directly for the base of ship! The astronauts clamber back into the ship, and fire their boosters at time t₀ when the boulder is falling at a vertical height of 10 km above the ledge.

Note that numerical answers are required for parts (a)-(c) and (e) below.

(2 points) (a) Find the angle ₀ and the acceleration of gravity g₀ on the planet.

(2 points) (b) Find the maximum height above the ledge reached by the boulder.

(2 points) (c) How much time t₂ does the team have left to escape when they fire their boosters at time t₀?

The astronauts use both their main and emergency boosters, which together (for times t>t₀) give the ship a total vertical acceleration (in m/sec²) of

a(t') = A(1 + e^(t'/)) ,

where = 5 sec and t' = t - t₀. Note that A already includes the planet's gravitational acceleration g₀.

(2 points) (d) Find an expression for the speed of the ship U(t') for t₂ > t' > 0 . You may leave your answer in terms of A, , and t'.

(2 points) (e) If the base of the ship must reach a height of 1 km above the point where the boulder strikes the ledge to avoid being destroyed by flying debris, find the minimum value of A that will allow the team to escape.