Problem 1

A small body is projected from level ground with initial speed

u

at angle

\theta

above the horizontal in a uniform gravitational field of magnitude

g

directed downward. Put the origin at launch, neglect air resistance and Earth curvature, and take

0<\theta<\pi/2

so the body rises and returns to its launch height. (a) Resolve the initial velocity and write the time-dependent motion. (b) Find the flight time

T

and maximum height

H

. (c) Derive horizontal range

R

and the angle that maximizes it for fixed

u

problems.proof.stepOf

\boxed{T=\frac{2u\sin\theta}{g},\quad H=\frac{u^2\sin^2\theta}{2g},\quad R=\frac{u^2\sin2\theta}{g};\quad R_{\max}=\frac{u^2}{g}\text{ at }\theta=\frac{\pi}{4}}

problems.proof.analysis

The requested quantities are collected with the condition for maximum range. The optimal angle assumes equal launch and landing heights.