2.1 Linear Motion
Linear motion is motion along a straight line. It is described using physical quantities such as distance, displacement, speed, velocity, and acceleration.
Key Quantities
- Distance ($s$): Total length of path traveled (Scalar, metre $\text{m}$).
- Displacement ($s$): Distance traveled in a specific direction / shortest distance between start and end point (Vector, metre $\text{m}$).
- Speed ($v$): Rate of change of distance ($v = \frac{s}{t}$, Scalar, $\text{m s}^{-1}$).
- Velocity ($v$): Rate of change of displacement ($v = \frac{s}{t}$, Vector, $\text{m s}^{-1}$).
- Acceleration ($a$): Rate of change of velocity ($a = \frac{v - u}{t}$, Vector, $\text{m s}^{-2}$).
Ticker Tape Analysis
A ticker timer operates at $50\text{ Hz}$, producing $50\text{ dots per second}$. The time interval between two adjacent dots is $1\text{ tick} = \frac{1}{50}\text{ s} = 0.02\text{ s}$.
- Equal spacing: Constant velocity ($a = 0$).
- Increasing spacing: Increasing velocity / Acceleration ($a > 0$).
- Decreasing spacing: Decreasing velocity / Deceleration ($a < 0$).
2.2 Linear Motion Graphs
Displacement-Time ($s$-$t$) Graph
- Gradient: Velocity ($v$).
- Horizontal line ($s$ constant): Object is stationary ($v = 0$).
- Straight sloped line: Uniform velocity.
- Curved line: Non-uniform velocity (acceleration).
Velocity-Time ($v$-$t$) Graph
- Gradient: Acceleration ($a$).
- Area under graph: Displacement ($s$).
- Horizontal line ($v$ constant): Uniform velocity ($a = 0$).
- Straight sloped line upwards: Uniform acceleration.
- Straight sloped line downwards: Uniform deceleration.
2.3 Equations of Linear Motion
For motion with uniform acceleration ($a$), four linear motion equations are applied:
- $v = u + at$
- $s = \frac{1}{2}(u + v)t$
- $s = ut + \frac{1}{2}at^2$
- $v^2 = u^2 + 2as$
Where: $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $s$ = displacement, $t$ = time interval.
2.4 Free Fall Motion and Gravitational Acceleration
- Free Fall: Motion of an object falling under the influence of gravity alone without air resistance.
- Gravitational Acceleration ($g$): Acceleration of a free-falling object ($g \approx 9.81\text{ m s}^{-2}$).
- Objects of different masses fall with the same acceleration ($g$) in a vacuum.
2.5 Inertia
Inertia is the natural tendency of an object to resist any change in its original state, whether at rest or in motion.
- Newton's First Law of Motion: An object remains at rest or continues to move with constant velocity unless acted upon by an external resultant force.
- Mass and Inertia: The inertia of an object depends only on its mass. A larger mass has greater inertia.
2.6 Momentum
Momentum ($p$) is the product of mass and velocity: $p = mv$ (Unit: $\text{kg m s}^{-1}$ or $\text{N s}$).
Principle of Conservation of Momentum
In a closed system with no external forces, total momentum before collision equals total momentum after collision.
- Elastic Collision: Objects separate after collision. Kinetic energy is conserved. Formula: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$.
- Inelastic Collision: Objects stick together after collision with common velocity $v$. Kinetic energy is NOT conserved. Formula: $m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$.
- Explosion: Objects start from rest ($u = 0$) and push apart. Formula: $0 = m_1 v_1 + m_2 v_2 \rightarrow m_1 v_1 = -m_2 v_2$.
2.7 Impulsive Force and Impulse
- Force ($F$): $F = ma = \frac{m(v-u)}{t}$ (Newton's Second Law: $F \propto \frac{\Delta p}{t}$).
- Impulse ($J$): Change in momentum, $J = F t = mv - mu$ (Unit: $\text{N s}$ or $\text{kg m s}^{-1}$).
- Impulsive Force ($F$): Rate of change of momentum during a collision ($F = \frac{mv - mu}{t}$).
- Impact time ($t$):
- Increasing impact time ($t\uparrow$) decreases impulsive force ($F\downarrow$) — e.g., airbags, crumpled zones, high-jump mats.
- Decreasing impact time ($t\downarrow$) increases impulsive force ($F\uparrow$) — e.g., hammer hitting a nail, karate chop.
2.8 Weight
Weight ($W$) is the gravitational force acting on an object: $W = mg$ (Unit: Newton, $\text{N}$). Weight is a vector quantity and changes depending on gravitational acceleration $g$.