Chapter Review
Motion and Dynamics
Motion and Graphs · Newton's Laws · Momentum and Impulse · Collisions · Projectile Motion
Motion Graphs: Slope and Area
The three motion graphs (s-t, v-t, a-t) are linked by differentiation (slope) and integration (area). Reading slopes and areas correctly lets you extract velocity, acceleration, and displacement from any graph.
Key Points
- •Slope of s-t graph = velocity; slope of v-t graph = acceleration
- •Area under v-t graph = displacement; area under a-t graph = change in velocity
- •Straight line on s-t → constant velocity; curve on s-t → acceleration present
- •Area below the time-axis on a v-t graph represents negative displacement (motion in reverse)
- •Use triangle (½bh), rectangle (bh), and trapezium (½(a+b)h) formulas for piecewise-linear v-t graphs
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 unknown you don't need.
Key Points
- •$v_f = v_i + at$ — use when displacement is not needed
- •$s = v_i t + \frac{1}{2}at^2$ — use when final velocity is unknown
- •$v_f^2 = v_i^2 + 2as$ — use when time is unknown
- •$s = \frac{v_i + v_f}{2} \times t$ — use when acceleration is unknown
- •For free fall, replace $a$ with $g ≈ 9.8$ m/s² (downward positive) or $-g$ (upward positive)
- •Distance in the $n^{th}$ second from rest follows the odd ratio 1 : 3 : 5 : 7…
Formula
$$s = v_i t + \frac{1}{2}at^2$$
Newton's Laws of Motion
The three laws form the foundation of classical mechanics: inertia preserves motion, net force produces acceleration, and every interaction generates equal-opposite force pairs on different bodies.
Key Points
- •First Law: $\sum F = 0 \implies a = 0$ — objects maintain their velocity unless acted on by an unbalanced force
- •Second Law: $F = ma$ — acceleration is proportional to net force and inversely proportional to mass
- •Third Law: $F_{AB} = -F_{BA}$ — action-reaction pairs always act on different objects and never cancel
- •Weight $W = mg$ is gravitational force; mass is intrinsic and location-independent
- •Valid only at speeds much less than $c$ (classical regime)
Formula
$$F = ma$$
Friction and Inclined Planes
Friction opposes relative motion and depends on the normal force and surface properties. On inclines, weight resolves into components parallel and perpendicular to the surface.
Key Points
- •Static friction adjusts up to $f_{s,max} = \mu_s N$; kinetic friction is constant at $f_k = \mu_k N$
- •$\mu_s > \mu_k$ — harder to start motion than to maintain it
- •Normal force on an incline: $N = mg\cos\theta$; driving component: $mg\sin\theta$
- •Critical angle for sliding: $\theta_c = \tan^{-1}(\mu_s)$
- •Friction does not depend on contact area — only on $\mu$ and $N$
Formula
$$a = g(\sin\theta - \mu_k \cos\theta)$$
Connected Bodies and Tension
For objects linked by massless, inextensible strings, find system acceleration first using total net force and total mass, then isolate individual bodies to find tension.
Key Points
- •System approach: $a = \frac{F_{net}}{m_{total}}$
- •Atwood machine: $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$, $T = \frac{2m_1 m_2 g}{m_1 + m_2}$
- •Tension is uniform throughout an ideal massless string
- •If $m_1 = m_2$ in an Atwood machine, $a = 0$ and $T = mg$ (equilibrium)
Formula
$$T = \frac{2m_1 m_2 g}{m_1 + m_2}$$
Momentum, Impulse, and Conservation
Momentum ($p = mv$) is conserved in all collisions when no external forces act. Impulse links force and time to the resulting momentum change, underpinning safety design.
Key Points
- •Impulse-momentum theorem: $F \Delta t = m(v_f - v_i) = \Delta p$
- •Conservation: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ in an isolated system
- •Increasing contact time reduces impact force for the same $\Delta p$ (airbags, crumple zones)
- •A bouncing object delivers impulse $2mv$ — double that of simply stopping
- •Recoil from rest: $m_1 v_1 = -m_2 v_2$ — lighter piece moves faster
Formula
$$J = F_{avg} \Delta t = \Delta p$$
Elastic Collisions
In elastic collisions both momentum and kinetic energy are conserved ($e = 1$). The relative velocity of approach equals the relative velocity of separation.
Key Points
- •Relative velocity shortcut: $v_1 - v_2 = v'_2 - v'_1$ (avoids quadratic KE equation)
- •Equal masses, one at rest: complete velocity exchange — first stops, second takes off
- •Light hits heavy ($m_1 \ll m_2$): light body reverses at ≈ same speed; heavy barely moves
- •Heavy hits light ($m_1 \gg m_2$): light body flies off at ≈ $2v_1$; heavy continues at ≈ $v_1$
- •General formula: $v'_1 = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2$
Formula
$$v'_1 = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2$$
Inelastic Collisions and Restitution
In inelastic collisions momentum is conserved but KE is not. The coefficient of restitution $e$ quantifies bounciness from 0 (stick together) to 1 (perfectly elastic).
Key Points
- •Perfectly inelastic: objects stick together, $v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$
- •KE lost: $\Delta KE = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(v_1 - v_2)^2$
- •Coefficient of restitution: $e = \frac{v'_2 - v'_1}{v_1 - v_2}$
- •From bounce heights: $e = \sqrt{h_2 / h_1}$
- •For partial inelastic problems, solve momentum conservation and $e$ equation simultaneously
Formula
$$e = \frac{v'_2 - v'_1}{v_1 - v_2}$$
Projectile Motion Fundamentals
A projectile moves in two independent directions: constant horizontal velocity ($a_x = 0$) and uniformly accelerated vertical motion ($a_y = g$). The resulting path is a parabola.
Key Points
- •Resolve initial velocity: $v_{ix} = v_i \cos\theta$ (constant), $v_{iy} = v_i \sin\theta$ (changes with $g$)
- •Time of flight: $T = \frac{2v_i \sin\theta}{g}$ (ground-to-ground only)
- •Maximum height: $H = \frac{v_i^2 \sin^2\theta}{2g}$
- •Range: $R = \frac{v_i^2 \sin 2\theta}{g}$; maximized at $\theta = 45°$
- •Horizontal launch from height $h$: fall time is $t = \sqrt{2h/g}$, independent of horizontal speed
Formula
$$R = \frac{v_i^2 \sin 2\theta}{g}$$
Projectile Symmetry and Special Results
Complementary launch angles produce equal ranges, speed is minimum at the peak, and the trajectory equation directly relates vertical to horizontal position.
Key Points
- •Complementary angles ($\theta$ and $90° - \theta$) give the same range but different heights and flight times
- •At the peak, speed is minimum: $V_{min} = v_i \cos\theta$ (only horizontal component survives)
- •Landing speed equals launch speed for same-level projectiles (only direction changes)
- •Trajectory equation: $y = x\tan\theta - \frac{gx^2}{2v_i^2 \cos^2\theta}$ — a downward-opening parabola
- •At $45°$: $R = 4H$ — a quick shortcut relating range to height
Formula
$$y = x\tan\theta - \frac{gx^2}{2v_i^2 \cos^2\theta}$$
Formulas
Equations of Motion (3rd)
Displacement without knowing final velocity.
Formula
$s = v_i t + \frac{1}{2}at^2$
Equations of Motion (4th)
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 proportional to normal force.
Formula
$f = \mu N$
Impulse-Momentum Theorem
Force over time equals change in momentum.
Formula
$F \Delta t = m(v_f - v_i)$
Conservation of Momentum
Total momentum conserved in isolated collisions.
Formula
$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$
Elastic Collision (v'₁)
Final velocity of body 1 in 1D elastic collision.
Formula
$v'_1 = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2$
Coefficient of Restitution
Ratio of separation speed to approach speed.
Formula
$e = \frac{v'_2 - v'_1}{v_1 - v_2}$
Horizontal Range
Range of a ground-to-ground projectile.
Formula
$R = \frac{v_i^2 \sin 2\theta}{g}$
Maximum Height
Peak height of a projectile.
Formula
$H = \frac{v_i^2 \sin^2\theta}{2g}$