Kinematics and Motion

Subject: Physics
Topic: 1
Cambridge Code: 0625

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Basic Concepts

Distance and Displacement

Distance - Total path length traveled (scalar)

• Can only increase
• No direction

Displacement - Straight-line change in position (vector)

• Can be positive or negative
• Has direction

Speed and Velocity

Speed - Rate of distance change (scalar) $v = \frac{\text{distance}}{\text{time}}$

Velocity - Rate of displacement change (vector) $v = \frac{\text{displacement}}{\text{time}}$

Average velocity: $\bar{v} = \frac{\text{total displacement}}{\text{total time}}$

Instantaneous velocity: $v = \frac{ds}{dt}$

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Acceleration

Acceleration - Rate of velocity change (vector) $a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t}$

Positive acceleration: Speed increasing (same direction as velocity) Negative acceleration (deceleration): Speed decreasing or direction change

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Equations of Motion

For uniform (constant) acceleration:

$v = u + at$

$s = ut + \frac{1}{2}at^2$

$v^2 = u^2 + 2as$

Where:

• u = initial velocity
• v = final velocity
• a = acceleration
• s = displacement
• t = time

Derivation

From definition of acceleration: $a = \frac{v - u}{t}$ $v = u + at$ ... (1)

From average velocity: $\bar{v} = \frac{u + v}{2}$ $s = \bar{v} \cdot t = \frac{u + v}{2} \cdot t$ ... (2)

Substituting (1) into (2): $s = \frac{u + (u + at)}{2} \cdot t = ut + \frac{1}{2}at^2$ ... (3)

From (1) and (3) eliminating t: $v^2 = u^2 + 2as$ ... (4)

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Velocity-Time Graphs

Interpreting Graphs

Gradient = acceleration Area under graph = displacement

Different Scenarios

Constant velocity:

• Horizontal line
• Zero acceleration
• Displacement = velocity × time

Constant acceleration:

• Straight line (non-zero gradient)
• Area is trapezoid or triangle
• Displacement = area

Changing acceleration:

• Curved line
• Non-constant gradient
• Area still equals displacement

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Displacement-Time Graphs

Interpreting Graphs

Gradient = velocity

Steeper slope = higher velocity

Horizontal line = stationary

Curved line = acceleration present

• Increasing slope: speeding up
• Decreasing slope: slowing down

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Free Fall

Free fall - Motion under gravity alone (ignoring air resistance) $a = g = 9.8 \text{ m/s}^2 \approx 10 \text{ m/s}^2$

Equations

Using standard equations with a = g:

$v = u + gt$ $s = ut + \frac{1}{2}gt^2$ $v^2 = u^2 + 2gs$

Special Cases

Dropped from rest: u = 0 $s = \frac{1}{2}gt^2$ $v = gt$

Thrown upward:

• Initial velocity u > 0 (upward)
• At maximum height: v = 0
• Returns to starting height: displacement = 0 (but time > 0)

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Projectile Motion

Projectile - Object under gravity with initial horizontal velocity

Horizontal and Vertical Motion

Horizontal:

• No acceleration
• Constant velocity
• $x = u_x t$ (where $u_x$ is initial horizontal velocity)

Vertical:

• Acceleration = g (downward)
• Changes with time
• $y = u_y t + \frac{1}{2}gt^2$ (where $u_y$ is initial vertical velocity)

Launch from Ground Level

Initial conditions:

• Horizontal velocity: $u_x = u\cos θ$ (where θ is launch angle)
• Vertical velocity: $u_y = u\sin θ$
• Initial height: 0

Range (horizontal distance): $R = \frac{u^2 \sin(2θ)}{g}$

Maximum height: $h = \frac{u^2 \sin^2(θ)}{2g}$

Time of flight: $t = \frac{2u\sin θ}{g}$

Launch from Height

Adjust initial height in vertical equation: $y = h_0 + u_y t - \frac{1}{2}gt^2$

Projectile lands when y = 0, solve for t

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Terminal Velocity

Terminal velocity - Maximum velocity reached when air resistance equals weight

$F_{\text{gravity}} = F_{\text{drag}}$ $mg = kv_t$ $v_t = \frac{mg}{k}$

Characteristics:

• Velocity constant (zero acceleration)
• Reached when net force = 0
• Depends on mass, shape, medium
• Occurs in gas or liquid

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Relative Velocity

Velocity of object A relative to B: $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

Example:

• Person on moving train
• Velocity relative to ground ≠ velocity relative to train

Vector Addition

For velocities at angles, use:

• Vector diagram (triangle rule)
• Components method
• Trigonometry

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Key Points

1. Distance vs displacement: Distance scalar, displacement vector
2. Speed vs velocity: Speed scalar, velocity vector
3. Acceleration = change in velocity / time
4. v² = u² + 2as useful when time not given
5. Free fall: a = g = 9.8 m/s²
6. Projectile motion: Horizontal and vertical independent
7. v-t graph gradient = acceleration, area = displacement
8. s-t graph gradient = velocity
9. Terminal velocity reached when air resistance = weight
10. Relative velocity depends on reference frame

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Practice Questions

1. Calculate velocities and accelerations
2. Use equations of motion
3. Analyze v-t graphs
4. Analyze s-t graphs
5. Solve free fall problems
6. Projectile motion calculations
7. Find range and maximum height
8. Analyze terminal velocity
9. Relative velocity problems
10. Multi-stage motion scenarios

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Revision Tips

• Draw diagrams for clarity
• Identify known and unknown
• Choose correct equation
• Check units always
• Be careful with signs (+ upward)
• Understand graph types
• Practice projectile motion
• Visualize motion mentally
• Use g = 10 m/s² for quick checks