A particle of mass � is projected from the ground with speed � at an angle � above the horizontal. Air…

Question

A particle of mass � is projected from the ground with speed � at an angle � above the horizontal. Air resistance is negligible and gravity is �. Derive expressions for the time of flight, maximum height, and horizontal range. Show that the range is maximum when �. Now suppose the particle is projected from a height � above the ground instead of from ground level. Derive a general expression for the time of flight. For the same initial speed �, find the angle � that maximizes the range when launched from height �.

ScanSolve AI Tutor · Accepted Answer

Here are the derivations for projectile motion:

Part 1: Particle projected from the ground

Let the initial speed be $v$ and the launch angle be $\theta$ above the horizontal. The acceleration due to gravity is $g$ .
The initial velocity components are:
Horizontal: $v_x = v \cos \theta$
Vertical: $v_y = v \sin \theta$

Time of Flight ( $T$ ) The particle starts at $y=0$ and returns to $y=0$ . Using the vertical displacement equation $y = v_y t - \frac{1}{2}gt^2$ :

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