Question

I NEED HELP WILL GIVE MEDAL ITS FREAKING HARD -_-;; A hot-air balloon is rising straight up at a constant speed of 7.1 m/s. When the balloon is 13.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 34.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them has the same altitude at the same time. How far above ground are these places? (Enter your answers from smallest to largest.)

Answer 1

On the assumption that you can ignore air resistance this becomes an application of the equations of motion.
For the balloon you have linear motion - so the height at any moment is given by s= s0 +ut (where s0 is the height at time 0)
For the pellet you have 'ballistic' motion - since the pellet is fired vertically upwards it has no horizontal velocity.
Its height at any moment is given by s=ut + 1/2 at^2
BUT the pellet doesn't start at the same time as the balloon.
The balloon has travelled 13m when the pellet is fired, so if we measure time from the moment the pellet is fired we know that s0= 13m

Therefore we have 2 equations:
height of balloon = 13+7.1*t
height of pellet = 34*t -9.81/2 *t^2

The question asks when these 2 heights are equal so the equation is

13+7.1*t=34*t-9.81/2 *t^2

This can be re-arranged to be a quadratic equation giving 2 solutions for t
(These 2 solutions are the time when the pellet passes the balloon on the way up and again on the way down).

Once you have the 2 values for t you can put them into the equation for the balloon (s= s0 +u*t) to find the height of the items at the times when they are equal.