A-Level Maths: Mechanics Practice

A-Level Maths — Mechanics Practice

18 MCQ practice problems covering core A-Level Mechanics content.

What These Questions Test

These problems test your ability to model real-world situations using mathematical mechanics. You will apply Newton’s laws, use SUVAT equations, analyse forces, and work with energy and momentum.

Typical question types:

• Kinematics: Using SUVAT equations for constant acceleration. Interpreting displacement-time, velocity-time, and acceleration-time graphs. Vertical motion under gravity.
• Forces and Newton’s Laws: Resolving forces into components. Applying $F = ma$ along a chosen axis. Friction ($F \leq \mu R$). Connected particles (pulleys, tow bars).
• Moments: Taking moments about a pivot ($M = Fd$). Conditions for equilibrium: $\sum F = 0$ and $\sum M = 0$. Centre of mass problems.
• Work, Energy, and Power: Work done $W = Fs\cos\theta$. Kinetic energy $E_{k} = \frac{1}{2}mv^{2}$. Potential energy $E_{p} = mgh$. Conservation of energy. Power $P = \frac{W}{t} = Fv$.
• Momentum: $p = mv$. Impulse $J = F\Delta t = \Delta p$. Conservation of momentum in collisions. Coefficient of restitution $e = \frac{\text{speed of separation}}{\text{speed of approach}}$.

Approach Strategy

1. Choose axes carefully. Align one axis with the acceleration (or the incline for slope problems). This simplifies the equations.
2. Draw a force diagram. For every object, identify all forces acting on it. Label magnitudes and directions.
3. Set up equations. Apply Newton’s second law along each axis: $\sum F_x = ma_x$ and $\sum F_y = ma_y$.
4. Solve simultaneously. For connected particles, write one equation for each particle, then eliminate the tension.

Intuition

Mechanics in maths is about translating physical situations into equations. The key skill is identifying the right mathematical model: SUVAT for kinematics, $F = ma$ for dynamics, $M = Fd$ for moments, and energy conservation for work problems.

Think of forces as “inputs” to a system. The net force determines the acceleration, which determines how the velocity changes, which determines how the position changes. It is a chain of cause and effect.

Worked Examples

Example 1: Connected Particles (Tow Bar)

Problem: Two particles A (2 kg) and B (3 kg) are connected by a light rigid tow bar. A horizontal force of 20 N is applied to A. Find the acceleration and the tension in the tow bar. (Assume no friction.)

Solution: Step 1: Draw force diagrams for each particle

• A: 20 N forward, tension $T$ backward
• B: tension $T$ forward

Step 2: Apply $F = ma$ to each particle

• A: $20 - T = 2a$ … (1)
• B: $T = 3a$ … (2)

Step 3: Substitute (2) into (1): $20 - 3a = 2a \Rightarrow 20 = 5a \Rightarrow a = 4 \, \text{m/s}^2$

Step 4: Find tension: $T = 3 \times 4 = 12 \, \text{N}$

Key insight: For connected particles, treat each particle separately, then link them through the tension.

Example 2: Projectile Motion

Problem: A ball is thrown horizontally from a cliff 45 m high with speed 12 m/s. Find the time of flight and the horizontal distance. ($g = 9.8 \, \text{m/s}^2$)

Solution: Step 1: Vertical motion (u = 0, a = g, s = 45): $s = ut + \frac{1}{2}at^2 \Rightarrow 45 = 0 + \frac{1}{2}(9.8)t^2$

Step 2: Solve for $t$: $t^2 = \frac{90}{9.8} = 9.184 \Rightarrow t = 3.03 \, \text{s}$ (to 3 s.f.)

Step 3: Horizontal motion (constant velocity): $d = vt = 12 \times 3.03 = 36.4 \, \text{m}$

Key insight: Horizontal and vertical motions are independent. The time of flight is determined entirely by the vertical motion.

Example 3: Work and Energy

Problem: A 5 kg block slides down a rough incline of length 8 m and angle 30° to the horizontal. The coefficient of friction is 0.2. Find the speed at the bottom. ($g = 9.8 \, \text{m/s}^2$)

Solution: Step 1: Calculate weight components:

• Along incline: $mg\sin 30° = 5 \times 9.8 \times 0.5 = 24.5 \, \text{N}$
• Perpendicular: $mg\cos 30° = 5 \times 9.8 \times \frac{\sqrt{3}}{2} = 42.44 \, \text{N}$

Step 2: Friction force: $F_f = \mu R = 0.2 \times 42.44 = 8.488 \, \text{N}$

Step 3: Work-energy theorem: Work done by gravity $-$ Work done by friction $= \Delta KE$ $24.5 \times 8 - 8.488 \times 8 = \frac{1}{2}(5)v^2$

Step 4: Solve: $196 - 67.9 = 2.5v^2 \Rightarrow 128.1 = 2.5v^2 \Rightarrow v = 7.16 \, \text{m/s}$

Key insight: Work-energy problems are often easier than using $F = ma$ because you don’t need to find acceleration first.

Common Mistakes

1. Using the wrong value of $g$. In A-Level mechanics, use $g = 9.8 \, \text{m/s}^{2}$ unless told otherwise. Some students use 10, which gives wrong answers.
2. Forgetting to resolve forces. When a force acts at an angle, you must split it into components before applying $F = ma$. A common error is using the full force magnitude in the equation of motion.
3. Confusing weight and mass. Weight is $W = mg$ (a force in newtons). Mass is $m$ (in kilograms). The equation $F = ma$ uses mass, not weight. If you use $F = mg + ma$ for an object on a slope, you have double-counted gravity.
4. Sign errors in vertical motion. If upward is positive, acceleration due to gravity is $a = -g$. Many students forget the negative sign, leading to incorrect results for projectile problems.

Cross-References

• Momentum: Mechanics practice covers momentum
• Energy and Work: Work-energy problems are tested
• Kinematics: Kinematics practice builds skills