How long after initially spotting the alien spaceship is it destroyed? What was the velocity of the spaceship when it was destroyed?

Question

A spaceship is 1500 meters above the Earth's surface descending at a velocity of 65m/s but slowing at a

rate of 4m/s^2. Worried about an alien attack, the US government spends 7 seconds debating the options

before firing a tactical nuclear missile at the spaceship. The missile is launched from a height of 125 meters

with an acceleration of 11m/s^2 (ignore the acceleration due to gravity in this problem, it's already been

factored into the objects' accelerations)

Answer 1

The problem statement seems incomplete, but I'll guess at some possible questions:

1. Does the tactical missile intercept the spaceship?
2. How far above earth's surface does the interception occur?

Both the spaceship and the missile are undergoing constant acceleration, so we can use the constant acceleration equations to find their positions.

spaceship: initial height = 1500 m, initial velocity -65 m/s, acceleration +4 m/s² (it's always good to draw a diagram, which I can't do here!)

so the spaceship's height is y = 1500 - 65*t +2t² m

missile: initial height = 125 m, initial velocity = 0 m/s, acceleration = + 11 m/s²

so the missile's height is y = 125 + 5.5*(t-7)² m - note that we use t-7, not t, because the missile is fired at t = 7 s!

Setting the two heights equal to each other, we can solve for t graphically or using the quadratic formula. Ignoring extraneous solutions (negative time solutions aren't physically relevant), we get t = 19.6 s. That answers question 1.

To find the height above earth's surface, we can plug this value into either of our height equations, giving us a height of 998 m above earth's surface.