Chapter Review

Motion and Force

Displacement, Velocity, Acceleration and Graphs · Newton Laws · Momentum and Impulse · Elastic and Inelastic Collisions · Projectile Motion

Displacement, Velocity and Acceleration

Displacement is the straight-line vector change in position from start to end, while distance is the total scalar path length traveled. Velocity is the rate of change of displacement; acceleration is the rate of change of velocity.

Key Points

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Displacement vs Distance: Displacement can be zero on a round trip while distance is always positive. Displacement is a vector; distance is a scalar.
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Average Speed ≠ Average Velocity: Speed = total distance / total time; velocity = displacement / time. For equal-distance trips at different speeds, use harmonic mean $\frac{2v_1 v_2}{v_1+v_2}$, not arithmetic mean.
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Instantaneous Velocity: Limit of $\Delta s/\Delta t$ as $\Delta t \to 0$; equals the tangent slope on an s-t graph.
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Acceleration Sign Convention: Same sign as velocity → speeding up; opposite signs → slowing down. Negative $a$ with negative $v$ still means speeding up.
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Dimensional Check: $[v] = LT^{-1}$, $[a] = LT^{-2}$. If your answer's units don't match, recheck your work.

Interpreting Motion Graphs

The three motion graphs (s-t, v-t, a-t) are linked by slope and area: slope of s-t gives velocity, slope of v-t gives acceleration, area under v-t gives displacement, area under a-t gives change in velocity.

Key Points

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s-t Graph: Gradient = velocity. Horizontal = stationary; straight diagonal = constant velocity; curve = accelerating/decelerating.
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v-t Graph: Gradient = acceleration; area = displacement. Positive area = forward; negative area = backward. Line crossing the time axis means direction reversal.
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a-t Graph: Area under curve = $\Delta v$. Constant $a$ = horizontal line → linear v-t graph → parabolic s-t graph.
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Total Distance vs Displacement: Distance = sum of absolute areas (all positive); displacement = algebraic sum with signs.
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Geometric Shapes: Triangle = $\frac{1}{2}bh$, rectangle = $bh$, trapezium = $\frac{1}{2}(a+b)h$ — these are essential for v-t graph problems.

Formula

$$v = \frac{ds}{dt}, \quad a = \frac{dv}{dt}, \quad \Delta s = \int v \, dt$$

Equations of Uniformly Accelerated Motion

Four kinematic equations relate displacement, velocity, acceleration, and time for constant acceleration along a straight line. Choose the equation that excludes the variable you don't need.

Key Points

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Missing $s$: $v_f = v_i + at$ — connects velocity, acceleration, and time directly.
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Missing $a$: $s = \frac{v_i + v_f}{2} t$ — displacement from average of initial and final velocities.
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Missing $v_f$: $s = v_i t + \frac{1}{2}at^2$ — displacement without knowing final velocity.
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Missing $t$: $v_f^2 = v_i^2 + 2as$ — final velocity without time (most useful for braking/falling problems).
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Nth Second: $s_n = v_i + \frac{a}{2}(2n-1)$ gives distance in a single specific second. From rest, successive seconds follow ratio 1:3:5:7...
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Free Fall: All objects fall at $g \approx 9.8$ m/s² regardless of mass. Acceleration remains $g$ even at the peak where $v = 0$.

Formula

$$v_f^2 = v_i^2 + 2as$$

Newton's Three Laws of Motion

First Law defines inertia and equilibrium (no net force → no acceleration). Second Law quantifies dynamics ($F = ma$). Third Law states that every action has an equal and opposite reaction acting on a different body.

Key Points

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First Law: $\sum F = 0 \implies a = 0$. Object at rest stays at rest; object in motion maintains constant velocity indefinitely.
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Inertia: Resistance to change in motion, measured by mass. More mass = harder to accelerate.
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Second Law: Acceleration direction matches net force direction. $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$. Weight $W = mg$ differs from mass.
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Third Law: $F_{AB} = -F_{BA}$ — action-reaction pairs act on DIFFERENT bodies simultaneously and never cancel on the same object.
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Normal Force Misconception: The normal force from a table on a book is NOT the reaction to its weight. The true reaction is the book's gravitational pull on Earth.

Formula

$$F = ma$$

Friction, Normal Force and Connected Bodies

Friction opposes relative motion between surfaces and is proportional to the normal force ($f = \mu N$). The normal force is the perpendicular contact force from a surface. Connected bodies linked by a light inextensible string share the same acceleration magnitude.

Key Points

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Static vs Kinetic: Static friction adjusts up to $\mu_s N$; kinetic friction is constant at $\mu_k N$. Always $\mu_s > \mu_k$.
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Normal Force on Inclines: $N = mg\cos\theta$. At $\theta = 0°$, $N = mg$; at $\theta = 90°$, $N = 0$.
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Critical Angle: Block slides when $mg\sin\theta > \mu_s mg\cos\theta$, giving $\theta_c = \tan^{-1}(\mu_s)$.
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No Area Dependence: Friction depends only on $\mu$ and $N$, not on contact surface area.
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Connected Bodies: System acceleration $a = F_{net}/m_{total}$, then isolate one body to find tension. Atwood machine: $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$.
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Car Propulsion: Static friction from road on tyres accelerates the car forward — friction enables motion, not just opposes it.

Formula

$$f = \mu N, \quad N = mg\cos\theta$$

Momentum and Impulse

Momentum ($p = mv$) is the vector quantity of motion. Impulse ($J = F\Delta t$) equals the change in momentum. In an isolated system with no external forces, total momentum is conserved during all collisions and explosions.

Key Points

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Momentum is a Vector: Always assign +/− signs based on a chosen positive direction before calculating.
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Impulse-Momentum Theorem: $J = F_{avg}\Delta t = \Delta p$. Same momentum change with longer contact time → smaller force (airbags, crumple zones).
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Unit Equivalence: $1 \text{ N·s} = 1 \text{ kg·m/s}$.
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Conservation: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ — holds for ALL collisions and explosions in isolated systems.
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Bouncing Doubles Impulse: An object bouncing back has $\Delta v = v_f + v_i$ (stops then reverses), giving $\Delta p = 2mv$ vs $mv$ for stopping.
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Explosions from Rest: Total $p = 0$, so $m_1v_1 = -m_2v_2$. Lighter fragment moves faster in the opposite direction.

Formula

$$J = F_{avg} \Delta t = m(v_f - v_i)$$

Elastic and Inelastic Collisions

Momentum is always conserved in collisions. In elastic collisions ($e = 1$), kinetic energy is also conserved and relative approach speed equals separation speed. In perfectly inelastic collisions ($e = 0$), objects stick together with maximum KE loss.

Key Points

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Elastic: Both momentum and KE conserved. Approach speed = separation speed: $v_1 - v_2 = v'_2 - v'_1$.
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Equal Masses, Elastic: Complete velocity exchange — incoming ball stops, target takes all velocity (Newton's cradle).
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Light Hits Heavy, Elastic: Light bounces back at ≈ original speed; heavy barely moves.
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Heavy Hits Light, Elastic: Heavy continues nearly unchanged; light flies off at ≈ $2v_1$, not $v_1$.
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Perfectly Inelastic: $v_f = \frac{m_1v_1 + m_2v_2}{m_1+m_2}$. KE lost = $\frac{1}{2}\frac{m_1m_2}{m_1+m_2}(v_1-v_2)^2$.
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Coefficient of Restitution: $e = \frac{v'_2-v'_1}{v_1-v_2}$. Range: 0 (stick) to 1 (perfect bounce). From heights: $e = \sqrt{h_2/h_1}$.

Formula

$$e = \frac{v'_2 - v'_1}{v_1 - v_2}$$

Elastic Collision Velocity Formulas

Combining momentum conservation with the relative velocity relation gives closed-form formulas for final velocities in 1D elastic collisions. Mass ratios determine whether bodies continue, reverse, or exchange velocities.

Key Points

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Body 1 Final Velocity: $v'_1 = \frac{m_1-m_2}{m_1+m_2}v_1 + \frac{2m_2}{m_1+m_2}v_2$. Sign of $(m_1-m_2)$ decides if body 1 continues or reverses.
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Body 2 Final Velocity: $v'_2 = \frac{2m_1}{m_1+m_2}v_1 + \frac{m_2-m_1}{m_1+m_2}v_2$. Body 2 always moves in the direction of the incoming object when $v_2 = 0$.
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Unit Shortcut: Masses appear as ratios — any consistent unit works (grams or kg cancel).
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Always Verify: After solving, check $m_1v'_1 + m_2v'_2 = m_1v_1 + m_2v_2$ and $v'_2 - v'_1 = v_1 - v_2$.

Formula

$$v'_1 = \frac{m_1-m_2}{m_1+m_2}v_1 + \frac{2m_2}{m_1+m_2}v_2$$

Projectile Motion

A projectile's horizontal and vertical motions are independent. Horizontal velocity stays constant (no horizontal force); vertical motion follows free-fall equations. The trajectory is a parabola.

Key Points

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Vector Resolution: $v_x = v_i\cos\theta$ (constant throughout); $v_y = v_i\sin\theta - gt$ (decreases going up, increases coming down).
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Time of Flight: $T = \frac{2v_i\sin\theta}{g}$ — depends only on vertical component. Time to peak = $T/2$.
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Maximum Height: $H = \frac{v_i^2\sin^2\theta}{2g}$. At peak, $v_y = 0$ but $v_x \neq 0$ — minimum speed = $v_i\cos\theta$.
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Range: $R = \frac{v_i^2\sin 2\theta}{g}$ — maximized at $\theta = 45°$. Complementary angles ($\theta$ and $90°-\theta$) give same range.
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At 45°: $R = 4H$ and $R_{max} = v_i^2/g$. Height uses $\sin^2\theta$; range uses $\sin 2\theta$ — don't confuse them.
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Horizontal Launch: Time from height $h$ is $t = \sqrt{2h/g}$ — do NOT use $T = 2v_i\sin\theta/g$ which is for oblique launches only.

Formula

$$R = \frac{v_i^2 \sin 2\theta}{g}, \quad H = \frac{v_i^2 \sin^2\theta}{2g}, \quad T = \frac{2v_i \sin\theta}{g}$$

Formulas

First Equation of Motion

Final velocity from initial velocity, acceleration, and time.

Formula

$$v_f = v_i + at$$

Fourth Equation of Motion

Final velocity without knowing time.

Formula

$$v_f^2 = v_i^2 + 2as$$

Newton's Second Law

Net force equals mass times acceleration.

Formula

$$F = ma$$

Friction Force

Friction is proportional to the normal force.

Formula

$$f = \mu N$$

Impulse-Momentum Theorem

Impulse equals change in momentum.

Formula

$$J = F_{avg} \Delta t = m(v_f - v_i)$$

Conservation of Momentum

Total momentum is conserved in an isolated system.

Formula

$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$

Coefficient of Restitution

Ratio of separation speed to approach speed (0 = stick, 1 = bounce).

Formula

$$e = \frac{v'_2 - v'_1}{v_1 - v_2}$$

Projectile Range, Height, and Time

Range uses sin2θ, height uses sin²θ, time uses sinθ.

Formula

$$R = \frac{v_i^2 \sin 2\theta}{g}, \quad H = \frac{v_i^2 \sin^2\theta}{2g}, \quad T = \frac{2v_i \sin\theta}{g}$$