Help me with Integrals?

Question

Two balls are thrown upwards from the edge of a cliff 432 ft above the ground. The first is thrown with an initial velocity of 48 ft/s and the other is thrown a second later with an initial velocity of 24 ft/s. DO the balls ever pass each other ?

Answer 1

For the first ball, the height s is given by

s= -16t^2+48t+432

For the second ball, since it is thrown one second later, the "initial time" is t=1 (as measured on the clock for the first ball), so:

s= -16(t-1)^2 +24(t-1)

Set the two equal to each other, and you have a nice messy quadratic equation to solve for t, which will give you the time or times at which the balls have the same height s. You then need to check whether the value or values of t you get make physical sense (t must be positive and the values of s must both be postive at time t.)

Answer 2

You are supposed to integrate the velocity functions of each ball over time to come up with the displacement functions. Then set the two position functions equal. The reason being that when the x(t) functions are equal, it means the balls are in the same position at the same time.