Question

A projectile has the same initial kinetic energy no matter what the angle of projection. Why doesn't it rise to the same maximum height in each case?

Answer 1

because according to energy conservation

loss in KE = gain PE

OR KE1 + PE1 = KE2 + PE2

=> KE1 - KE2 = PE2

1/2mV^2 - 1/2m(Vcos(theta))^2 = mgHmax

So clearly Max height is fuction of theta , and KEintital = KE1 = 1/2mV^2 is independent of theta

Answer 2

ANSWER :

Maximum height depends on the vertical component of initial velocity, u,  of projection. 

If angle of  projection is ?  with the horizontal , then,

vertical component of initial velocity = u sin ? 

So, 

Total K.E. initially = 1/2 m u^2 (independent of ?) 

K.E. in the vertical direction = 1/2 m (u sin ?)^2 

K.E. in the horizontal direction = 1/2 m (u cos ?)^2

At maximum height, K.E = 0  (since velocity in the vertical direction is 0) 

but it gains P.E. by gaining the height h. 

So,

Gain of P.E = Loss of K.E.

=> m g h = 1/2  m (u sin ?)^2 

=> h = (u sin ? )^2  / 2g  = (u^2 / 2g) sin^2 ?  [ u^2 / 2g is a constant for the given value of u]

Hence, h is a function of ?. If ? varies, sin ? varies and thereby h too varies.

This explains why, h will be different for different angles of projection.  (ANSWER).