Physics

Problem

For a projectile launched from level ground with initial speed

v_0

, two different angles

\theta

and

(90^\circ-\theta)

give the same horizontal range. Let

H_1

and

H_2

be the corresponding maximum heights reached for these two angles. a. (a) Obtain expressions for

H_1

and

H_2

in terms of

v_0

\theta

and

g

. b. (b) Prove that

H_1 + H_2 = \dfrac{v_0^2}{2g}

, showing that the sum of the maximum heights is independent of

\theta

. c. (c) Show that the product of the maximum heights is given by

H_1 H_2 = \dfrac{v_0^4 \sin^2 2\theta}{16 g^2}

. d. (d) Hence determine the value of

\theta

for which the product

H_1 H_2

is maximum and evaluate this maximum product in terms of

v_0

and

g