Kinematics and projectile motion: study notes

1. Position, displacement, velocity and acceleration

Position is where a body is relative to a chosen origin. Displacement $s$ is the change in position: a vector, so it has a direction and can be negative or zero even after a long journey. Distance is the scalar length of the path actually travelled, so it never decreases. A ball thrown up and caught at the same point has zero displacement but a non-zero distance.

Velocity is the rate of change of position, $v = \dfrac{\Delta s}{\Delta t}$, a vector; speed is the rate of change of distance, a scalar. Acceleration is the rate of change of velocity, $a = \dfrac{\Delta v}{\Delta t}$, also a vector, so a body slowing down or changing direction is accelerating.

Average values use the whole interval: average velocity $= \dfrac{\text{total displacement}}{\text{total time}}$, average speed $= \dfrac{\text{total distance}}{\text{total time}}$. Instantaneous values apply at one moment and are found by shrinking the interval to zero, which on a graph means the gradient of the tangent at that instant. Always state the sign convention you are using (for example, upwards positive) before you write any equation.

2. Determining quantities from graphs

These graph rules are not in the booklet and must be learned.

Displacement-time graph: the gradient is velocity. A straight line means constant velocity; a curve of increasing steepness means increasing speed; a horizontal section means the body is at rest. The gradient of the tangent gives the instantaneous velocity; the gradient of the chord between two points gives the average velocity over that interval.

Velocity-time graph: the gradient is acceleration and the area between the line and the time axis is displacement. Area below the axis counts as negative displacement, so total distance adds the magnitudes while displacement subtracts them. A straight sloping line means uniform acceleration; a curve means non-uniform acceleration.

Acceleration-time graph: the area under the line is the change in velocity.

For a body thrown vertically upwards with upwards positive, the velocity-time graph is a straight line of gradient $-g$ that crosses the axis at the highest point, and the displacement-time graph is an inverted parabola.

3. Equations of motion for uniform acceleration

When acceleration is constant the four equations in the data booklet apply, with $u$ the initial velocity, $v$ the final velocity, $a$ the acceleration, $s$ the displacement and $t$ the time:

$$s = \frac{(u+v)t}{2}, \quad v = u + at, \quad s = ut + \frac{1}{2}at^{2}, \quad v^{2} = u^{2} + 2as$$

Each equation omits one of the five quantities, so list what you know and what you want, then choose the equation that avoids the unknown you do not need. Every symbol is measured from the same origin and in the same positive direction, so a displacement that ends below the start is negative if upwards is positive.

Non-uniform acceleration (a changing gradient on a velocity-time graph, for example a falling object with air resistance) cannot be handled with these equations. In that case use graphical methods: tangent gradients for instantaneous values and areas, estimated by counting squares, for displacement.

Free fall near the Earth's surface is the standard case of uniform acceleration: $a = g = 9.81$ m s$^{-2}$ downwards, independent of mass when air resistance is ignored.

4. Projectile motion without fluid resistance

A projectile is any body moving only under gravity after launch. The key idea is to treat the horizontal and vertical motions separately, linked only by the common time $t$.

Horizontal: no force, so no acceleration; the horizontal velocity stays at $u_{x} = u\cos\theta$ and range $= u_{x}t$.

Vertical: constant acceleration $g$ downwards; the initial vertical velocity is $u_{y} = u\sin\theta$ and all four equations apply with $a = -9.81$ m s$^{-2}$ if upwards is positive.

Three launch cases appear. Launched horizontally from a height $h$: $u_{y} = 0$, so $h = \frac{1}{2}gt^{2}$ fixes the time of flight, then the range follows from $u_{x}t$. Launched above the horizontal: the body rises until $v_{y} = 0$, then falls; the path is a symmetric parabola if it lands at launch height. Launched below the horizontal: $u_{y}$ is negative from the start and the time of flight is shorter than for a horizontal or upward launch at the same speed from the same height.

At any instant the speed is $\sqrt{v_{x}^{2} + v_{y}^{2}}$ and the direction is $\tan\phi = v_{y}/v_{x}$. At the highest point the velocity is purely horizontal (not zero) and the acceleration is still $g$. A body dropped and one projected horizontally from the same height hit the ground at the same time; a body thrown upwards and one thrown downwards at the same speed from the same height hit the ground with the same speed but at different times.

5. Qualitative effect of fluid resistance

Air (or any fluid) exerts a drag force opposite to the velocity that grows with speed. Consequences to quote in explain parts: the horizontal velocity now decreases, so the range is shorter; the maximum height is lower; the trajectory is no longer a symmetric parabola but steeper on the descent; the acceleration is no longer constant or purely vertical; the speed on landing is lower than it would be without drag (and, for a projectile that returns to launch height, lower than the launch speed); and, for a long fall, drag increases until it balances weight, the acceleration becomes zero and the body falls at a constant terminal speed. For a vertical fall, time of flight is longer than without resistance. Each of these is far more likely to be credited if you name the cause: a resistive force opposing motion whose size increases with speed.

Formulas

Worked examples

Can you...

• Define displacement, velocity and acceleration as vectors, and explain the difference between average and instantaneous values?
• Read velocity from the gradient of a displacement-time graph, and acceleration and displacement from the gradient and area of a velocity-time graph?
• Choose the correct equation of motion from the booklet and apply a consistent sign convention when the motion ends below the start?
• Resolve a launch velocity into $u\cos\theta$ and $u\sin\theta$, use the vertical motion to find the time of flight and the horizontal motion to find the range, and recombine components for the impact speed and angle?
• Explain, with reference to a drag force that increases with speed, the effect of air resistance on range, height, trajectory, acceleration, impact speed and terminal speed?