Projectile Motion & Simple Harmonic Motion Notes

Last reviewed 16 Sept 2026 · 4 min read

Projectile motion

A projectile is a body thrown into the air and moving under gravity alone (air resistance neglected). Its motion combines:

uniform velocity horizontally — no horizontal force, and
uniformly accelerated motion vertically — acceleration $g$ downward.

Let the body be projected with velocity $u$ at angle $α$ above the horizontal.

FormulaProjectile on level ground

Components: $u_{x} = u cos α$ (constant), $u_{y} = u sin α$
Time of flight: $T = \frac{2 u sin α}{g}$
Maximum height: $H = \frac{u ^{2} sin ^{2} α}{2 g}$
Horizontal range: $R = \frac{u ^{2} sin 2 α}{g}$
Maximum range at $α = 45°$ : $R_{ma x} = \frac{u ^{2}}{g}$ (and then $H = R_{ma x} /4$ )
Equation of trajectory (a parabola): $y = x tan α - \frac{g x ^{2}}{2 u ^{2} cos ^{2} α}$

RememberComplementary angles

Angles $α$ and $90° - α$ give the same range (since $sin 2 α$ is equal), but different heights and times of flight. For range $R$ and time of flight $T$ : $tan α = \frac{g T ^{2}}{2 R}$ . Also $tan α = 4 H / R$ .

Velocity at any instant: $v_{x} = u cos α$ , $v_{y} = u sin α - g t$ ; at the highest point the velocity is horizontal and equals $u cos α$ .

Projectile fired horizontally from a height $h$

t = \frac{2 h}{g}, range = u \frac{2 h}{g}, v_{s t r ik e} = u^{2} + 2 g h

Projectile on an inclined plane

For a plane inclined at $β$ , projection at $α$ to the horizontal ( $α > β$ ), range up the plane:

R = \frac{2 u ^{2} cos α sin ( α - β )}{g cos ^{2} β}, R_{ma x} = \frac{u ^{2}}{g ( 1 + sin β )} when α = 45° + \frac{β}{2}

Simple harmonic motion (SHM)

A body performs SHM when its acceleration is proportional to its displacement from a fixed point (the mean position) and always directed towards that point:

a = - ω^{2} x

It is the projection on a diameter of uniform circular motion with angular velocity $ω$ .

FormulaSHM relations (amplitude

A

)

Displacement: $x = A sin ω t$
Velocity: $v = ω A^{2} - x^{2}$ ; maximum $v_{ma x} = ω A$ at the mean position
Acceleration: $a = - ω^{2} x$ ; maximum $a_{ma x} = ω^{2} A$ at the extreme positions
Period $T = \frac{2 π}{ω} = 2 π \frac{displacement}{acceleration}$ ; frequency $f = 1/ T$
Energy: total $= \frac{1}{2} m ω^{2} A^{2}$ (constant); KE is maximum at the mean position, PE at the extremes.

Spring–mass system

A mass $m$ on a spring of stiffness $k$ :

ω = \frac{k}{m}, T = 2 π \frac{m}{k} = 2 π \frac{δ _{s t}}{g}

$δ_{s t}$ = static deflection $m g / k$ . Natural frequency $f_{n} = \frac{1}{2 π} \frac{k}{m}$ . Springs in series reduce $k$ (longer period); in parallel increase $k$ .

Simple pendulum

A point mass on a light string of length $L$ , small oscillations:

T = 2 π \frac{L}{g}

Period is independent of the mass and (for small amplitudes) of the amplitude.
A seconds pendulum has $T$ = 2 s → $L \approx 0.994$ m at $g$ = 9.81 m/s².

Compound pendulum

A rigid body swinging about a horizontal axis at distance $h$ from its CG, radius of gyration $k_{G}$ :

T = 2 π \frac{k _{G}^{2} + h ^{2}}{g h}

Equivalent simple pendulum length $L = \frac{k _{G}^{2} + h ^{2}}{h}$ . The period is minimum when $h = k_{G}$ .

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