Dynamics: Motion, Projectiles, Forces & Energy

TAKEAWAYS:

• Motion is described by displacement, velocity, and acceleration vectors (including projectile motion’s parabolic path).
• Newton’s laws explain forces: $F=ma$, action-reaction pairs, and equilibrium ($\sum F=0$).
• Energy conservation links kinetic ($KE=\frac{1}{2}m{v}^2$), potential ($PE=mgh$), and work ($W=F\cdot d$).
• Projectiles follow symmetric trajectories: time-of-flight depends on vertical motion, range on horizontal velocity.
• Collisions are elastic (energy conserved) or inelastic (momentum conserved: $m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$).
• Exam focus: Solve numerical problems using equations, draw motion/force diagrams, and interpret graphs.

1. Kinematics in Two and Three Dimensions

Key Concepts

• Displacement (s): Vector from start to finish (units: meters, m).
• Velocity (v): Rate of displacement change ($\vec{v}=\frac{\Delta \vec{s}}{\Delta t}$).
• Acceleration (a): Rate of velocity change ($\vec{a}=\frac{\Delta \vec{v}}{\Delta t}$).
• Projectile motion: Combines horizontal (constant velocity) and vertical (accelerated by gravity, $g=9.8{\text{m/s}}^2$) motion.

Equations of Motion (for constant acceleration)

1. $v=u+at$
2. $s=ut+\frac{1}{2}a{t}^2$
3. $v^2=u^2+2as$

For projectiles:

• Horizontal motion: $x=v_{0x}t$ (no acceleration).
• Vertical motion: $y=v_{0y}t-\frac{1}{2}g{t}^2$.

Worked Example 1: Projectile Motion

A ball is kicked with initial velocity $20\text{m/s}$ at $30^{\circ }$ to the horizontal. Find:

1. Time of flight
2. Maximum height
3. Horizontal range

Solution:

1. Resolve velocity: $v_{0x}=20\cos 30^{\circ }=20\times \frac{\sqrt{3}}{2}=10\sqrt{3}\text{m/s}$ $v_{0y}=20\sin 30^{\circ }=10\text{m/s}$

2. Time of flight: At landing, $y=0$. Use $y=v_{0y}t-\frac{1}{2}g{t}^2$: $0=10t-4.9t^2$ $t(10-4.9t)=0$ $t=0$ (start) or $t=\frac{10}{4.9}\approx 2.04\text{s}$.

3. Maximum height: At peak, $v_y=0$. Use $v=u+at$: $0=10-9.8t\Rightarrow t=1.02\text{s}$. Substitute into $y=v_{0y}t-\frac{1}{2}g{t}^2$: $y=10\times 1.02-4.9\times (1.02{)}^2\approx 5.1\text{m}$.

4. Horizontal range: $x=v_{0x}\times t_{\text{flight}}=10\sqrt{3}\times 2.04\approx 35.4\text{m}$.

2. Newton’s Laws of Motion

Key Concepts

Worked Example 2: Forces and Acceleration

A 5 kg box is pushed with a 20 N force. Friction opposes motion with 5 N. Find the acceleration.

Solution:

1. Draw a free-body diagram (see below).
2. Net force: $F_{\text{net}}=20\text{N}-5\text{N}=15\text{N}$.
3. Use $F=ma$: $a=\frac{F_{\text{net}}}{m}=\frac{15}{5}=3{\text{m/s}}^2$.

3. Work, Energy, and Power

Key Concepts

• Work (W): Force applied over a distance ($W=\vec{F}\cdot \vec{d}$).
• Kinetic Energy (KE): Energy of motion ($KE=\frac{1}{2}m{v}^2$).
• Potential Energy (PE): Stored energy (gravitational: $PE=mgh$).
• Conservation of Energy: $K{E}_i+P{E}_i=K{E}_f+P{E}_f$ (no friction).
• Power (P): Rate of work ($P=\frac{W}{t}$).

Worked Example 3: Energy Conservation

A 2 kg ball is dropped from 10 m. Find its speed just before hitting the ground.

Solution:

1. Initial energy: $P{E}_i=mgh=2\times 9.8\times 10=196\text{J}$.
2. Final energy: $K{E}_f=\frac{1}{2}m{v}^2$.
3. By conservation: $196=\frac{1}{2}\times 2\times v^2$.
4. Solve for $v$: $v^2=196\Rightarrow v=14\text{m/s}$.

4. Collisions

Key Concepts

Worked Example 4: Inelastic Collision

A 3 kg cart moving at 4 m/s collides with a 2 kg cart at rest. They stick together. Find the final velocity.

Solution:

1. Initial momentum: $p_i=3\times 4+2\times 0=12\text{kg⋅m/s}$.
2. Final momentum: $p_f=(3+2)v_f=5v_f$.
3. By conservation: $12=5v_f\Rightarrow v_f=2.4\text{m/s}$.

5. Applications of Dynamics

flowchart TD
    A["Satellite in Orbit"] --> B["Centripetal Force"]
    B --> C["F = mv²/r"]
    C --> D["Gravitational Force"]
    D --> E["F = GMm/r²"]
    E --> F["Orbital Velocity v = sqrt(GM/r)"]

Real-World Examples

1. Sports: Cricket ball trajectories, golf swings (projectile motion).
2. Engineering: Car safety (crumple zones absorb energy in collisions).
3. Astronomy: Satellite orbits (centripetal force $F=\frac{m{v}^2}{r}$).
4. Medicine: Blood flow in arteries (Bernoulli’s principle).

Exam Tip

1. Diagrams are mandatory: Always draw free-body diagrams or motion paths.
2. Units matter: Answer in m/s, m/s², or J (not cm/s).
3. Projectile questions: Break into horizontal/vertical components.
4. Energy problems: Use conservation ($K{E}_i+P{E}_i=K{E}_f+P{E}_f$).
5. Collision questions: Check if KE is conserved (elastic/inelastic).
6. Graphs: Interpret $v$-$t$ or $a$-$t$ graphs for displacement/velocity.

NEB-Style Questions

Short Answer (5 marks)

1. A stone is thrown horizontally from a cliff 20 m high with speed 10 m/s. Find:
   • Time to reach the ground.
   • Horizontal distance traveled.
   • Vertical velocity just before impact.

Long Answer (10 marks)

2. A 1000 kg car moving at 20 m/s brakes to a stop in 5 s.
   • Calculate the braking force.
   • If the car’s KE is converted to heat, how much energy is dissipated?
   • Draw a $v$-$t$ graph and shade the area representing distance.

Conceptual (3 marks)

3. Explain why a projectile’s trajectory is symmetric. What happens if air resistance is considered?

Note: Practice NEB past papers (2075–2080) for question patterns. Focus on numerical problems and diagram-based questions!