Equations of motion

Kinematics is the study of how objects move. In this topic, we focus on motion in a straight line (one dimension), using key quantities and equations to describe and predict motion.

Key Quantities

• Distance: How much ground an object has covered (scalar).
• Displacement: The straight-line change in position from start to end (vector).
• Speed: How fast an object moves (scalar).
• Velocity: The rate of change of displacement (vector).
• Acceleration: The rate of change of velocity (vector).

Definitions and Relationships

• Speed: $speed = \frac{distance}{time}$
• Velocity: $velocity = \frac{displacement}{time}$
• Acceleration: $acceleration = \frac{change\ in\ velocity}{time}$

Typical SI units:

• Distance/displacement: metre (m)
• Speed/velocity: metre per second (m s$^{-1}$)
• Acceleration: metre per second squared (m s$^{-2}$)

Graphical Representation

• Displacement–time graph: Slope (gradient) gives velocity.
• Velocity–time graph: Slope gives acceleration; area under the graph gives displacement.

Determining Quantities from Graphs

• Displacement from velocity–time graph: Area under the curve.
• Velocity from displacement–time graph: Gradient of the curve.
• Acceleration from velocity–time graph: Gradient of the curve.

Equations of Uniformly Accelerated Motion

When acceleration is constant, the following equations apply (for motion in a straight line):

Let:

• $u$ = initial velocity
• $v$ = final velocity
• $a$ = acceleration
• $s$ = displacement
• $t$ = time

These are called the "equations of motion" or "SUVAT equations".

Derivation of Equations

The equations come from the definitions of velocity and acceleration, assuming acceleration is constant.

• From $a = \frac{v-u}{t}$, rearrange for $v$ to get $v = u + at$.
• Displacement is area under velocity–time graph (trapezium area), leading to $s = \frac{(u+v)}{2} t$.
• Substitute $v$ from $v = u + at$ into the above to get $s = ut + \frac{1}{2}at^2$.
• Rearranging and eliminating $t$ gives $v^2 = u^2 + 2as$.

Free Fall

Objects falling freely near Earth's surface (ignoring air resistance) have constant acceleration $g \approx 9.81$ m s$^{-2}$ downward.

• Use the equations of motion with $a = g$ (downward).

Experimental Determination of $g$

A simple experiment to determine acceleration due to gravity:

• Drop a steel ball from rest and measure the time $t$ it takes to fall a known height $h$.
• Use $s = \frac{1}{2}gt^2$ (since $u = 0$).
• Rearranged: $g = \frac{2s}{t^2}$

Motion in Perpendicular Directions

If an object moves with constant velocity in one direction (e.g., horizontal) and constant acceleration in a perpendicular direction (e.g., vertical), the motions can be analysed separately.

• Horizontal motion: $x = ut$ (if $a = 0$)
• Vertical motion: $y = ut + \frac{1}{2}at^2$ (with $a = g$ for free fall)

This principle is used in projectile motion.

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Summary:

• Distinguish between distance/displacement, speed/velocity, and acceleration.
• Use graphs to interpret and calculate motion.
• Apply equations of motion for constant acceleration, including free fall.
• Understand and describe experiments to measure $g$.
• Analyse motion in perpendicular directions independently.