Find the time at which the two balls collide

[Guess78]

A ball is thrown straight up from the ground with speed Vo . At the same instant, a second ball is dropped from rest from a height H , directly above the point where the first ball was thrown upward. There is no air resistance.


1-Find the time at which the two balls collide.
Express your answer in terms of the variables "Vo","H" , and appropriate constants..


2-Find the value of "H" in terms of "Vo" and "g" so that at the instant when the balls collide, the first ball is at the highest point of its motion.
Express your answer in terms of the variables "Vo" and "g" .

 

[LowlyPion]

Welcome to PF.

How would you think to start?

 

[nlightner]

1- To find the time at which the two balls collide, we can use the equation of motion for the first ball thrown upwards:
h = h0 + v0t - 1/2gt^2
Where h is the height of the first ball at any time t, h0 is the initial height (in this case, 0 since it is thrown from the ground), v0 is the initial velocity (given by Vo), g is the acceleration due to gravity (9.8 m/s^2), and t is the time.
Similarly, for the second ball dropped from rest, the equation of motion is:
h = h0 - 1/2gt^2
Where h is the height of the second ball at any time t, h0 is the initial height (given by H), g is the acceleration due to gravity, and t is the time.
Since we want to find the time at which the two balls collide, we can set the two equations equal to each other and solve for t:
h0 + v0t - 1/2gt^2 = h0 - 1/2gt^2
Simplifying, we get:
v0t = 0
Therefore, t = 0
This means that the two balls will collide at the same time they are released, or t = 0.

2- To find the value of H in terms of Vo and g so that at the instant when the balls collide, the first ball is at the highest point of its motion, we can use the same equation of motion for the first ball:
h = h0 + v0t - 1/2gt^2
At the highest point of its motion, the velocity of the first ball will be 0, since it is momentarily at rest before falling back down. Therefore, we can set v0t = 0 and solve for h0:
0 = h0 - 1/2gt^2
h0 = 1/2gt^2
Substituting t = 0 (since the balls collide at this time), we get:
h0 = 0
This means that H, the initial height of the second ball, must also be 0 for the first ball to be at its highest point of motion at the instant of collision. Therefore, H = 0.