Force, Motion & Gravitation

Describing motion

• Rest and motion are relative: an object is in motion if its position changes with time compared with a reference point.
• Distance is the total length of the path travelled (scalar). Displacement is the shortest straight-line distance from the initial to the final position, with direction (vector). Distance ≥ |displacement|; they are equal for straight-line motion in one direction.
• Speed = distance / time (scalar). Velocity = displacement / time (vector).
• Average speed = total distance / total time. If an object covers equal distances at speeds $v_1$ and $v_2$, the average speed is $\frac{2v_1{v}_2}{v_1+v_2}$ (the harmonic mean), not the simple average.
• Acceleration = change in velocity / time; unit m/s². Retardation (deceleration) is negative acceleration.
• Uniform motion: equal distances in equal times (constant velocity). Non-uniform motion: velocity changes.

Equations of motion (constant acceleration)

where $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $t$ = time, $s$ = displacement. The distance covered in the $n$-th second is $s_n=u+\frac{a}{2}(2n-1)$.

Graphs

• Distance–time graph: the slope gives speed. A straight line inclined upward = uniform speed; a horizontal line = at rest.
• Velocity–time graph: the slope gives acceleration; the area under the graph gives displacement. A horizontal line = uniform velocity; an inclined straight line = uniform acceleration.

Free fall

Near the Earth, all bodies fall with the same acceleration $g\approx 9.8{\text{m/s}}^2$ (in the absence of air resistance) — Galileo's result, shown by the feather-and-coin experiment in a vacuum (Newton's tube). For a body dropped from rest: $v=gt$, $h=\frac{1}{2}g{t}^2$. For a body thrown upward with speed $u$: maximum height $H=\frac{u^2}{2g}$; time of ascent $=u/g$; total time of flight $=2u/g$.

Circular motion

In uniform circular motion the speed is constant but the velocity changes direction, so the body is accelerating. The centripetal acceleration is $a=\frac{v^2}{r}$ directed to the centre, and the centripetal force is $F=\frac{m{v}^2}{r}$. Centrifugal force is the apparent outward force felt in a rotating frame. Examples of centripetal force: the tension in a string, gravitation for planets, friction for a car on a bend.

Projectile motion

A projectile launched at angle $\theta$ with speed $u$ has: time of flight $T=\frac{2u\sin \theta }{g}$, maximum height $H=\frac{u^2{\sin }^2\theta }{2g}$, range $R=\frac{u^2\sin 2\theta }{g}$. The range is maximum at 45° and is the same for angles $\theta$ and $90°-\theta$.