I’m a retired (66) photographer who has had a life-long interest in mathematics. However, my brain doesn't seem wired for math.

I’m working my way through the Morris Kline book. I have a copy of Maple 10 to assist me, but I’m interested in understanding what I’m doing not just obtaining an answer.

Here is my question:

A body is thrown into the air with an initial velocity V ft./sec. What initial velocity is required to double the maximum height previously obtained? I got a very clumsy looking ½(4s +at2)/t. His answer is a neat sqrt2V ft./sec. After several hours of trying to solve the problem I have no idea how he obtained the answer.

 

If you measure upwards from the ground then V is positive, the acceleration a is negative and the displacement s is given by

s = V t +1/2 a t2

where t is time measures from the instant that the body is released. The body returns to the ground when s = 0, so

V t +1/2 a t2 = 0

Solving for t gives t = o and t = -2V/a so the body is at its maximum height at t =  -V/a and this maximum height is

smax = s( -V/a ) = - V2/2a 

The question is then, what initial velocity will give a velocity of V at height smax, for that velocity will allow the body to rise another smax before starting to fall. Let U be this initial velocity. With this initial velocity, the velocity of the body at time t is given by U + at and the displacement by s = U t +1/2 a t2. Let t0 be the time when the body reaches a height of smax and then you want

V = U + a t0 and  - V2/2a = U t0 +1/2 a t20

I then used the first equation to write t0 = (V - U)/a  substituted this value into the second equation and solved for U. This gave me U = √2 V.

Penny