Displacement, Velocity and Acceleration: MDCAT Physics notes
Displacement, Velocity and Acceleration for MDCAT: distance vs displacement, average and instantaneous velocity, relative velocity, acceleration.
Distance and displacement
Distance is the total length of the path actually travelled. It is a scalar (magnitude only) and can never decrease.
Displacement is the shortest straight-line distance from the initial to the final position, in the direction of the final position. It is a vector (magnitude and direction). Its SI unit is the metre.
Because a straight line is the shortest path between two points, $|\text{displacement}| \le \text{distance}$, so
$$\frac{\text{displacement}}{\text{distance}} \le 1$$
The two are equal only for motion in a straight line without turning back. After a complete circle (or any closed path) the displacement is zero while the distance is not.
| Motion | Distance | Displacement |
|---|---|---|
| Half circle of radius $r$ | $\pi r$ | $2r$ (the diameter) |
| Full circle of radius $r$ | $2\pi r$ | 0 |
| Quarter circle of radius $r$ | $\pi r/2$ | $\sqrt2\,r$ |
Worked example: a runner goes 400 m east and then 100 m west. Distance = 500 m; displacement = 300 m east. Displacement : distance = 3 : 5.
The displacement of a body at distance $r$ from a reference point (for example the Sun measured from the Earth) is simply $r$, directed along the line joining them.
Velocity
Velocity is the rate of change of displacement. It is a vector; speed is its scalar magnitude.
$$\vec v_{av} = \frac{\Delta \vec d}{\Delta t}, \qquad \vec v_{ins} = \lim_{\Delta t\to 0}\frac{\Delta \vec d}{\Delta t}$$
- Average velocity depends only on the net displacement and the total time. Two trips of the same duration that end at the same point have the same average velocity, however different their paths. A trip that returns to its start has zero average velocity.
- Instantaneous velocity is the velocity at a particular instant. On a curved path it is directed along the tangent to the path at that point.
- In one dimension velocity may be positive or negative; the sign simply shows direction.
- Uniform velocity: equal displacements in equal intervals of time, however small. Variable velocity: unequal displacements in equal intervals of time, or a change of direction.
Worked example: a body is displaced 24 m in 8 s. $v_{av} = 24/8 = 3\ \mathrm{m\,s^{-1}}$.
Relative velocity
The velocity of A relative to B is $\vec v_{AB} = \vec v_A - \vec v_B$.
- Same direction: subtract the speeds.
- Opposite directions: add the speeds.
- A passenger sitting in a moving car has zero velocity relative to the car, whatever the car's speed relative to the road.
Worked example: trains at 50 km/h and 40 km/h. Same direction: relative velocity 10 km/h. Opposite directions: 90 km/h.
Acceleration
Acceleration is the time rate of change of velocity:
$$\vec a = \frac{\Delta \vec v}{\Delta t}, \qquad \text{unit: } \mathrm{m\,s^{-2}}$$
It is a vector. It exists whenever speed or direction changes, so a body moving in a circle at constant speed is accelerating. When acceleration is opposite to velocity the body slows down (deceleration).
Key formulas
- $v_{av} = \dfrac{\text{displacement}}{\text{time}}$; average speed $= \dfrac{\text{distance}}{\text{time}}$
- $v_{AB} = v_A - v_B$
- $a = \dfrac{v_f - v_i}{t}$
- 1 km/h $= \tfrac{5}{18}\ \mathrm{m\,s^{-1}}$ (so 72 km/h = 20 m/s)
Common MDCAT traps
- Displacement/distance can equal 1 but can never exceed it.
- For a semicircle, distance : displacement = $\pi : 2$, not $\pi : 1$.
- Variable velocity is about unequal displacements, not unequal distances.
- Vehicles moving in opposite directions: add the speeds.
- Instantaneous velocity is tangential, not along the radius.
Quick revision
- Distance: scalar; displacement: vector.
- Closed path: displacement zero.
- Average velocity uses net displacement.
- Acceleration = rate of change of velocity.