The four suvat equations only apply when acceleration is constant. List s, u, v, a, t, tick the three you know, then pick the equation without the unknown you don't need.
Graphs: gradient of a displacement–time graph = velocity; gradient of a velocity–time graph = acceleration; area under a velocity–time graph = displacement; area under an acceleration–time graph = change in velocity.
Average velocity \(=\dfrac{\Delta s}{\Delta t}\); instantaneous velocity is the gradient of the tangent at a point.
Projectiles: treat horizontal and vertical motion separately. Horizontal velocity is constant (no air resistance); vertical acceleration is \(g=9.8\text{ m s}^{-2}\) downwards. Time links the two.
Launch at speed \(u\) and angle \(\theta\): \(u_x=u\cos\theta\), \(u_y=u\sin\theta\). The range is greatest at 45° on level ground.
Air resistance reduces the maximum height and range and makes the path asymmetric: the descent is steeper than the ascent.
Fluid resistance increases with speed, so a falling object reaches terminal velocity when drag equals weight (resultant force zero).
Worked example
A ball is thrown horizontally at 12 m s⁻¹ from the top of a 20 m cliff. Ignoring air resistance, find the time of flight and how far from the cliff it lands.
Vertical: \(u_y=0,\ s=20\text{ m},\ a=9.8\text{ m s}^{-2}\), so \(20=\tfrac12(9.8)t^{2}\).
Draw a free-body diagram for one object at a time: weight, normal force, tension, friction, drag, buoyancy, each as an arrow from the object.
Equilibrium: resultant force is zero (at rest or constant velocity). Resolve forces into perpendicular components; on a slope, weight has components \(mg\sin\theta\) along and \(mg\cos\theta\) perpendicular to the surface.
Third-law pairs act on different objects, so they never cancel each other out.
Conservation of momentum: in a closed system with no external force, total momentum before = total momentum after. Applies to collisions and explosions.
Elastic collisions conserve kinetic energy; inelastic collisions do not (energy goes to heat, sound, deformation). Momentum is conserved in both.
Increasing collision time (airbags, crumple zones) reduces the average force for the same change in momentum.
In uniform circular motion speed is constant but velocity changes direction, so there is an acceleration towards the centre. The centripetal force is provided by a real force (tension, friction, gravity).
Worked example
A 2.0 kg trolley moving at 3.0 m s⁻¹ hits a stationary 1.0 kg trolley and they stick together. Find their common velocity and the kinetic energy lost.
Momentum before \(=2.0\times3.0=6.0\text{ kg m s}^{-1}\).
\(6.0=3.0\,v\Rightarrow v=\mathbf{2.0\text{ m s}^{-1}}\).
\(E_k\) before \(=\tfrac12(2.0)(3.0)^{2}=9.0\text{ J}\); after \(=\tfrac12(3.0)(2.0)^{2}=6.0\text{ J}\).
Kinetic energy lost \(=\mathbf{3.0\text{ J}}\): the collision is inelastic.
Exam tip: Momentum is a vector: choose a positive direction first and give velocities in the opposite direction a minus sign.
A.3Work, energy and power
Key terms
Work done (W)
Energy transferred by a force: W = Fs cos θ, in J.
Kinetic energy
Energy of motion: ½mv².
Gravitational potential energy
Energy stored due to position in a gravitational field: mgΔh near Earth.
Elastic potential energy
Energy stored in a stretched or compressed object: ½kx².
Power
Rate of doing work or transferring energy, in W (J s⁻¹).
Efficiency
Useful energy (or power) out ÷ total energy (or power) in.
A rigid body is in equilibrium when both the resultant force and the resultant torque are zero. Take moments about the point that removes the most unknowns.
A couple is a pair of equal, opposite, non-collinear forces: it produces a torque but no resultant force.
Moment of inertia depends on how mass is distributed about the axis: mass further away gives a larger \(I\).
Rolling without slipping: \(v=\omega r\). Total kinetic energy \(=\tfrac12mv^{2}+\tfrac12I\omega^{2}\), so objects with larger \(I\) roll down a slope more slowly.
Conservation of angular momentum: a skater who pulls in their arms decreases \(I\), so \(\omega\) increases.
Worked example
A flywheel with I = 0.50 kg m² starts from rest and has a constant torque of 2.0 N m applied for 5.0 s. Find its final angular velocity and rotational kinetic energy.
\(\alpha=\dfrac{\tau}{I}=\dfrac{2.0}{0.50}=4.0\text{ rad s}^{-2}\).
\(\omega=\alpha t=4.0\times5.0=\mathbf{20\text{ rad s}^{-1}}\).
Postulates of special relativity: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum is the same for all inertial observers.
Galilean transformations work at everyday speeds but predict that light speed depends on the observer, which experiments contradict.
Time dilation: a moving clock runs slow. Length contraction: a moving object is shorter along its direction of motion.
Simultaneity is relative: events simultaneous in one frame need not be simultaneous in another.
Evidence: muons made in the upper atmosphere reach the ground in large numbers even though their proper lifetime (about 2.2 μs) is too short without time dilation.
Spacetime diagrams: worldlines of light are at 45°; the axes of a moving frame are tilted towards the light line by angle \(\theta\) with \(\tan\theta=v/c\).
Worked example
A muon with proper lifetime 2.2 μs moves at 0.98c. Find its lifetime measured in the Earth frame and the distance it travels in that time.