Motion, Forces, Energy & Graphs: Exam-focused Summary

Basic Concepts of Motion

• Displacement: the vector distance and direction from a reference point; SI unit: m.

• Velocity: rate of change of displacement; a vector; SI unit: m s^{-1}; average velocity = total displacement ÷ total time.

• Speed: rate of change of distance; magnitude of velocity; average speed may differ from average velocity if motion reverses direction.

Equations of Motion for Uniform Acceleration

• Constant-acceleration equations (u = initial, v = final, a = acceleration, s = displacement, t = time):

  • $v = u + a t$

  • $v^2 = u^2 + 2 a s$

  • $s = ut + \tfrac{1}{2} a t^2$

  • $s = \tfrac{1}{2} (u + v) t$

• Signs can be positive or negative for displacement (s), velocity (u, v), and acceleration (a).

• Approach to problems (summary):
  1) verify constant acceleration; 2) note knowns (u, v, a, s, t); 3) decide which quantity to find; 4) choose the equation containing knowns and required quantity; 5) substitute and solve; 6) check reasonableness.

Vertical Motion under Gravity

• Gravity acceleration: $g = 9.81\ \mathrm{m\,s^{-2}}$ downward.

• Sign convention: choose upwards as positive; acceleration is $-g$ when up is positive.

• Key results: moving upward slows until velocity zero; then falls.

• Useful equations (vertical):

  • $v^2 = u^2 + 2 a s$ with a = -g for upward-positive convention.

  • When starting from rest, $s = \tfrac{1}{2} a t^2$.

• Maximum height (v = 0): $H = \dfrac{u^2}{2g}$ (for vertical launch).

Projectiles – Motion in Two Dimensions under Gravity

• Start with speed U at angle (\theta) to horizontal.

• Components:

  • Horizontal: $U<em>x = U\cos\theta$, acceleration $a</em>x = 0$ (constant velocity)

  • Vertical: $U<em>y = U\sin\theta$, acceleration $a</em>y = -g$

• Key times and heights:

  • Time to reach max height: $t<em>{\text{up}} = \dfrac{U</em>y}{g} = \dfrac{U\sin\theta}{g}$

  • Maximum height: $H = \dfrac{U_y^2}{2g} = \dfrac{(U\sin\theta)^2}{2g}$

  • Horizontal range (landing at same height): $R = \dfrac{U^2 \sin(2\theta)}{g}$

• Velocity at any time is the vector with components $v<em>x = U\cos\theta$ and $v</em>y = U\sin\theta - g t$.

• Resultant velocity at impact: $|\mathbf{v}| = \sqrt{v<em>x^2 + v</em>y^2}$.

Graphs of Motion

• Displacement-time (d-t) graph:

  • Slope = velocity; a straight line => constant velocity; curved => changing velocity.

• Velocity-time (v-t) graph:

  • Gradient = acceleration; straight line with constant gradient => constant acceleration; horizontal line => zero acceleration (constant velocity).

  • Area under v-t graph (for the time interval) = displacement.

• Instantaneous velocity from a curved d-t: use the gradient of the tangent at the desired point.

• Exam tips:

  • When calculating gradients, use long segments for accuracy.

  • Use tangents carefully to find instantaneous velocity.

Velocity as a Vector and Signs

• Velocity is a vector: magnitude and direction.

• In one-dimensional motion, forward vs backward is represented by positive/negative sign.

• Displacement and velocity signs indicate direction relative to chosen axis.

Vectors: Addition and Resolution

• Resultant of two vectors: draw vectors head-to-tail and connect tail to final head; or use components.

• Components: for a vector (V) making angle (\phi) with a chosen axis,

  • Horizontal: $V_x = V \cos\phi$

  • Vertical: $V_y = V \sin\phi$

• For two vectors, parallelogram rule or triangle rule can be used; for three or more, use a vector polygon or sum components.

Free-Body Diagrams and Newton's Laws

• Free-body diagram shows all forces acting on a body:

  • Weight: $W = mg$ (acts downward through the centre of mass)

  • Normal reaction: $N$ (perpendicular to contact surface)

  • Friction: $F_f$ (along contact surface; direction opposes motion or tendency to move)

  • Tension/Thrust: along strings or springs

  • Air resistance: opposite to motion

• Newton's Laws:

  • First Law (equilibrium): a body at rest or moving with constant velocity has net force zero.

  • Second Law: vector form, $\mathbf{F}_{\text{net}} = m \mathbf{a}$. For components along a given direction: sum of forces in that direction = m a along that direction.

  • Third Law: action-reaction pairs are equal in magnitude and opposite in direction; act on different bodies.

• Problem-solving approach: draw FBD, resolve forces along the direction of acceleration, apply Newton's laws, and solve.

Forces on an Incline and Equilibrium Problems

• Resolve forces parallel and perpendicular to the incline:

  • Parallel: sum = 0 gives friction if in limiting equilibrium or motion.

  • Perpendicular: normal force balances weight component perpendicular to plane.

• Friction: static friction up to a maximum value; direction depends on whether tendency is to move up or down the plane.

Energy, Work, and Power

• Work: $W = F s \cos\theta$ for constant forces; for a non-constant force, use integral or area under a force-distance graph.

• Kinetic Energy: $KE = \tfrac{1}{2} m v^2$

• Gravitational Potential Energy: $GPE = m g h$

• Work-Energy Principle (mechanical energy): if no non-conservative forces, total mechanical energy is conserved:

  • Initial: $KE<em>i + GPE</em>i$; Final: $KE<em>f + GPE</em>f$; set equal to solve for unknowns.

• Work-Energy Theorem: when non-conservative forces are present, work done by these forces equals the change in kinetic energy.

• Power: $P = \frac{dW}{dt} = F v$ (instantaneous: (\mathbf{P} = \mathbf{F} \cdot \mathbf{v})).

Specific Energy Considerations and Practice Tips

• Energy lost to friction manifests as heat (and sometimes sound).

• In vertical motion with air resistance, energy transfer includes work done against air; total mechanical energy not strictly conserved.

• Ferris wheel example: energy exchange between kinetic and gravitational potential energy; some energy dissipated as heat.

Stress, Strain, and the Young Modulus (Materials Topic)

• Stress: $\sigma = \dfrac{F}{A}$ (force per unit cross-sectional area).

• Strain: $\varepsilon = \dfrac{\Delta L}{L_0}$ (extension per original length).

• Young's Modulus: $E = \dfrac{\sigma}{\varepsilon} = \dfrac{F L_0}{A \Delta L}$

• Elastic region obeys Hooke's law; yield point, UTS (ultimate tensile stress), and breaking point are key material properties.

• Ductile vs brittle: ability to undergo plastic deformation before breaking vs sudden fracture.

Exam Techniques (Brief Tips)

• Always define quantities clearly and use vector directions (positive/negative) consistently.

• For graphs, identify whether you are dealing with d-t or v-t graphs and apply the correct interpretation (slope vs area vs gradient).

• When asked for a resultant of multiple forces, draw free-body diagrams and resolve into two perpendicular directions to sum components.

• In energy problems, decide whether conservation of energy applies (no non-conservative forces) or whether work-energy must be used (non-conservative forces present).

Quick Reference Formulas

• Kinematics (uniform acceleration):

  • $v = u + a t$

  • $v^2 = u^2 + 2 a s$

  • $s = ut + \tfrac{1}{2} a t^2$

  • $s = \tfrac{1}{2} (u + v) t$

• Projectile components: $U<em>x = U\cos\theta, { } U</em>y = U\sin\theta$; $R = \dfrac{U^2 \sin 2\theta}{g}$

• Gravity: $GPE = m g h,\ ext{KE} = \tfrac{1}{2} m v^2$

• Work: $W = F s \cos\theta$; Power: $P = F v = \dfrac{W}{t}$

• Stress/Strain: $\sigma = \dfrac{F}{A},\ \varepsilon = \dfrac{\Delta L}{L_0},\ E = \dfrac{\sigma}{\varepsilon}$

• Gravity constant: $g \approx 9.81\ \mathrm{m\,s^{-2}}$

• Sign conventions: always state which direction is positive for displacements, velocities, and accelerations.

Title: Motion and Energy – Exam-focused Summary for Quick Review