Kinematics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Kinematics — Key Facts

Kinematics describes motion without considering its causes. The fundamental quantities are displacement (s), velocity (v), and acceleration (a). All ECAT questions use the SI system: metres (m), seconds (s), and metres per second (m/s).

Equations of Motion (Rectilinear Motion with Constant Acceleration):

• $v = u + at$
• $s = ut + \frac{1}{2}at^2$
• $v^2 = u^2 + 2as$
• $s = \frac{(u+v)}{2}t$

where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

Key Definitions:

• Speed: rate of change of distance (scalar, always positive)
• Velocity: rate of change of displacement (vector, can be positive or negative)
• Acceleration: rate of change of velocity; positive a means speeding up in the direction of motion

⚡ ECAT Exam Tip: In ECAT 2024, 2–3 questions came from kinematics. Always check the sign of acceleration — if a and v have the same sign, object speeds up; if opposite signs, it slows down. For objects thrown upward, a = -g = -9.8 m/s².

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Projectile Motion (Motion in 2D):

When an object is launched at an angle θ with initial speed u, its motion splits into:

• Horizontal component: $u_x = u \cos\theta$ (constant, no acceleration)
• Vertical component: $u_y = u \sin\theta$ (acceleration = -g)

Key Quantities:

• Time of flight: $T = \frac{2u\sin\theta}{g}$
• Maximum height: $H = \frac{u^2\sin^2\theta}{2g}$
• Horizontal range: $R = \frac{u^2\sin 2\theta}{g}$

Special Cases:

• Range is maximum when θ = 45°
• Complementary angles (θ and 90°-θ) give the same range but different trajectories
• At the highest point, velocity is purely horizontal: v_y = 0, v_x = u cosθ

Relative Velocity: If object A moves with velocity v_A and B moves with v_B, then velocity of A relative to B: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

⚡ ECAT Exam Tip: For river-boat problems — effective speed downstream = v_boat + v_river; upstream = v_boat - v_river. Time to cross a river of width d is always t = d/v_boat (perpendicular component is unchanged).

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for complete mastery…

Derivation of Projectile Motion Equations:

Consider an object launched from origin with velocity u at angle θ:

Position at time t:

• x(t) = (u cosθ)t
• y(t) = (u sinθ)t - ½gt²

Eliminating t to get trajectory: $t = \frac{x}{u\cos\theta}$

Substituting: $y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$

This is the equation of a parabola — confirming projectile motion is parabolic.

For Objects Dropped from Height:

If an object is thrown horizontally from height h:

• x(t) = ut (where u is horizontal speed)
• y(t) = h - ½gt²
• Time to ground: $t = \sqrt{\frac{2h}{g}}$
• Horizontal range: $R = u\sqrt{\frac{2h}{g}}$

Uniform Circular Motion:

When an object moves in a circle of radius r with constant speed v:

• Angular displacement: $\theta = \frac{s}{r}$
• Angular velocity: $\omega = \frac{d\theta}{dt} = \frac{v}{r}$
• Time period: $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$
• Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$

Non-Uniform Circular Motion:

When speed changes:

• Tangential acceleration: $a_t = \frac{dv}{dt}$
• Total acceleration: $\vec{a} = a_r \hat{r} + a_t \hat{\theta}$
• Radial (centripetal) component: $a_r = \frac{v^2}{r}$ toward centre

⚡ ECAT 2023 Question Pattern: Recent ECAT papers emphasise problem-solving. Focus on: (1) identifying given quantities, (2) choosing correct equation, (3) consistent sign convention. Banking of roads and conical pendulum are also in the syllabus.

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