Advanced mechanics review

Use this page to check the whole module. Read the summary, make sure you can derive each result in the equation list, then do the test without looking back.

Module summary

• 5A Projectile motion. Assumptions (constant g, no air resistance). Horizontal and vertical motion are independent: $u_x$ constant, $a_y = -g$. Resolve $u_x = u\cos\theta$, $u_y = u\sin\theta$. Horizontal launch: time set by the drop. Final velocity from components.
• 5B Deriving the equations. $t = \frac{2u\sin\theta}{g}$, $H = \frac{u^2\sin^2\theta}{2g}$, $R = \frac{u^2\sin2\theta}{g}$ (level ground). 45° maximum; complementary angles equal range. From a height: solve the quadratic. $v = \sqrt{u^2 + 2gh}$ regardless of angle. Air resistance lowers range and best angle.
• 5C Projectile practical. Ramp, bench, carbon paper or video/strobe. Linearise: $y$ against $t^2$ (gradient $g/2$); range against launch speed. Validity, reliability, accuracy; random vs systematic errors.
• 5D Uniform circular motion. $v = 2\pi r/T$, $\omega = \Delta\theta/t$, $v = r\omega$, $a_c = v^2/r$, $F_c = mv^2/r$. Centripetal force is provided by a real force. Tangential path when released. Whirling stopper: $F$ against $v^2$ has gradient $m/r$.
• 5E Banked tracks and loops. Flat: $v_{\max} = \sqrt{\mu g r}$. Banked design speed: $\tan\theta = v^2/rg$; friction down the slope if faster, up if slower. Conical pendulum. Loops: $N = mg + mv^2/r$ (bottom), $N = mv^2/r - mg$ (top), $v_{\min} = \sqrt{gr}$.
• 5F Work, energy and torque. Centripetal force does no work (perpendicular). Energy conservation in vertical circles. $\tau = r_\perp F = rF\sin\theta$ in N m. Balanced torques.
• 5G Gravitational fields. $F = GMm/r^2$ (third-law pair, $r$ between centres). $g = GM/r^2$, inverse square law. Field lines. $g$ varies with latitude, altitude, geology; GRACE.
• 5H Satellites and orbits. Newton's cannon. $v = \sqrt{GM/r}$, $T = 2\pi\sqrt{r^3/GM}$, mass cancels. LEO (~90 min) vs geostationary (equatorial, 35 786 km, 1 day). Apparent weightlessness is free fall. Orbital decay.
• 5I Kepler's laws. Ellipses with the Sun at a focus; equal areas in equal times (fastest at perihelion; K and U trade, E constant); $r^3/T^2 = GM/4\pi^2$ for finding central mass or comparing orbits.
• 5J Gravitational energy. $U = -GMm/r$, zero at infinity. Circular orbit: $K = GMm/2r$, $E = -GMm/2r$. Higher orbit: more E, less K. $v_{\text{esc}} = \sqrt{2GM/r}$ (11.2 km/s for Earth).

Putting it together

The whole module is Newton's second law applied to curved paths. In projectile motion the force (gravity) is constant in size and direction, so the path is a parabola, and we split the motion into horizontal and vertical parts. In circular motion the net force is constant in size but always points to the centre, so it changes the direction of motion but never the speed. Orbits join the two ideas: a satellite is a projectile moving so fast that gravity, which now changes direction as it goes around the planet, acts as a centripetal force. Equating $\frac{GMm}{r^2}$ with $\frac{mv^2}{r}$ gives orbital speed, period and Kepler's third law in one line.

Energy runs alongside. Gravity does no work in a circular orbit, but it does in a projectile's flight, in a vertical loop and around an ellipse, trading kinetic for potential energy. Using $U = -GMm/r$ instead of $mgh$ lets you handle any distance, including the extreme case of escape.

Key equations and where they come from

Some of these appear on the formulae sheet and some don't, but you can be asked to derive any of them.

• Range $R = \frac{u^2\sin2\theta}{g}$: set $\Delta y = 0$ to get the time of flight, then multiply by $u_x$.
• Maximum height $H = \frac{u^2\sin^2\theta}{2g}$: use $v_y^2 = u_y^2 + 2a_y\Delta y$ with $v_y = 0$.
• Banked design speed $\tan\theta = \frac{v^2}{rg}$: divide $N\sin\theta = \frac{mv^2}{r}$ by $N\cos\theta = mg$.
• Orbital speed $v = \sqrt{\frac{GM}{r}}$: set $\frac{GMm}{r^2} = \frac{mv^2}{r}$.
• Kepler's third law $\frac{r^3}{T^2} = \frac{GM}{4\pi^2}$: substitute $v = \frac{2\pi r}{T}$ into the orbital speed result.
• Orbital energy $E = -\frac{GMm}{2r}$: add $K = \frac{GMm}{2r}$ to $U = -\frac{GMm}{r}$.
• Escape velocity $v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$: set $\frac12mv^2 - \frac{GMm}{r} = 0$.

Key words

angular velocity · aphelion · apparent weight · banked track · centripetal acceleration · centripetal force · complementary angles · conical pendulum · design speed · eccentricity · escape velocity · field line · free fall · geostationary orbit · gravitational field strength · gravitational potential energy · inverse square law · Kepler's laws · linearise · low Earth orbit · orbital decay · perihelion · period · projectile · range · reliability · semi-major axis · systematic error · time of flight · torque · trajectory · uniform circular motion · validity