physics help?

Question

Suppose that Galileo is in the Leaning Tower of Pisa. He throws a cannonball straight down from height H above the ground. The ball leaves his hands with an initial speed Vo. At the same instant, his friend throws a cannonball straight up from ground level with an initial speed of 2Vo. When the cannonballs collide, they are traveling in the same direction and Galileo's cannonball is
traveling with 7 times the speed of his friend's. Ignore air resistance.

a. At what time does the collision occur? Express your answer in terms of H and Vo
only.
b. At what height above the ground does the collision occur? Express your answer in terms of H only

Answer 1

The collision happens at height y at time t. At any given time t, Galileo's ball will have a height of H - Vo*t - 0.5*g*t^2 and his pal's ball will have a height of 2Vo*t - 0.5*g*t^2, where g is 9.8 m/s^2. Those formulae should look familiar to you. Set the two heights equal to each other, and you'll see that the t^2 terms cancel. It will be easy to solve for t in terms of H and Vo.

Solving for y is tougher. At any given time t, Galileo's ball has a velocity of -Vo - g*t and his pal's will have a velocity of 2Vo - g*t. Set the first equal to 7 times the second, and solve for t to get a value for t in terms of Vo instead of H. Then, use either of the height formulas from above and sub in for t. If you use the first solution for t in the Vo*t term, and the product of both solutions for the t^2 term, you should have a solution in terms of H only.