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

problems.proof.prevStep problems.proof.nextStep

x(t)=u\cos\theta\,t,\quad y(t)=u\sin\theta\,t-\frac12gt^2

problems.proof.analysis

Integrating gravitational acceleration gives uniform horizontal motion and accelerated vertical motion. The projectile returns to its launch height and experiences no drag.