Linear Motion & Force

Syllabus reference

Unit 2, Topic 1 — 25 hours (including practicals)

Vectors and scalars

Scalars have magnitude only (e.g. speed, distance, mass, energy). Vectors have both magnitude and direction (e.g. velocity, displacement, force, momentum).

Vectors can be symbolised as F, F̃ or F⃗.

To find the resultant vector in one dimension, assign a positive direction and add algebraically.

Linear motion concepts

Displacement (\(s\)): The change in position of an object — a vector quantity (m).
Velocity (\(v\)): The rate of change of displacement — a vector quantity (m s⁻¹).
Acceleration (\(a\)): The rate of change of velocity — a vector quantity (m s⁻²).

Average velocity is calculated over an entire journey. Instantaneous velocity is the velocity at a specific moment in time (the gradient of a displacement–time graph at that point).

Interpreting motion graphs

Graph	Gradient gives	Area under gives
Displacement–time	Velocity	—
Velocity–time	Acceleration	Displacement
Acceleration–time	—	Change in velocity

Use minimum and maximum lines of best fit to determine the uncertainty of the gradient.

Equations of motion

For uniformly accelerated motion in one dimension (constant acceleration):

SUVAT equations

[ v = u + at ] [ s = ut + \tfrac{1}{2}at^2 ] [ v^2 = u^2 + 2as ]

where: \(s\) = displacement, \(u\) = initial velocity, \(v\) = final velocity, \(a\) = acceleration, \(t\) = time

Worked example: Free fall

Question: A ball is dropped from rest from a height of 20.0 m. How long does it take to hit the ground? (\(g = 9.8\) m s⁻²)

Solution: \(u = 0\), \(s = 20.0\) m, \(a = 9.8\) m s⁻²

[ s = ut + \tfrac{1}{2}at^2 \implies 20.0 = 0 + \tfrac{1}{2}(9.8)t^2 ] [ t^2 = \frac{2 \times 20.0}{9.8} = 4.08 \implies t = 2.02 \text{ s} ]

Mandatory practical: Acceleration due to gravity

Conduct an experiment to verify the value of \(g = 9.8\) m s⁻² on Earth's surface. Linearise a non-linear dataset (e.g. plot \(t^2\) vs \(s\)) and calculate the equation of the linear trend line.

Newton's laws

First law (inertia): An object remains at rest or in uniform motion unless acted upon by an unbalanced force.

Second law: \(F_{net} = ma\), or equivalently \(a = \frac{F_{net}}{m}\).

Third law: For every action there is an equal and opposite reaction.

Free-body diagrams

Construct diagrams showing all forces acting on an object: weight (\(F_g = mg\)), normal force (\(F_N\)), tension (\(T\)), friction (\(f\)), drag, and applied forces.

Momentum and impulse

Momentum is the product of mass and velocity:

Key formulas

\[ p = mv \]

Impulse = change in momentum = area under a force–time graph:

\[ \text{Impulse} = F\Delta t = \Delta p = mv - mu \]

Conservation of momentum (in the absence of external forces):

\[ \sum mv_{before} = \sum mv_{after} \]

Worked example: Collision

Question: A 2.0 kg trolley moving at 3.0 m s⁻¹ collides with a stationary 1.0 kg trolley. They stick together. Find the final velocity.

Solution:

[ m_1 v_1 + m_2 v_2 = (m_1 + m_2)v_f ] [ (2.0)(3.0) + (1.0)(0) = (3.0)v_f \implies v_f = 2.0 \text{ m s}^{-1} ]

Energy

Key formulas

Work done: \(W = Fs\) (force × displacement in direction of force)

Kinetic energy: \(E_k = \tfrac{1}{2}mv^2\)

Gravitational potential energy: \(E_p = mgh\)

The work–energy theorem: the net work done on an object equals its change in kinetic energy.

Elastic collision — both momentum and kinetic energy are conserved.

Inelastic collision — momentum is conserved, but kinetic energy is not (some is converted to heat, sound, deformation).

Simulations and videos

PhET Simulations:

Crash Course Physics:

External resources: