Projectile motion resolved into independent horizontal and vertical components, Newton's law of universal gravitation, and the gravitational field around a mass.

What this key area is asking

The SQA wants you to treat projectile motion as two independent motions (constant-velocity horizontal and accelerating vertical), apply Newton's law of universal gravitation to find the force between two masses, and describe the gravitational field around a mass.

Projectile motion

To launch at an angle $\theta$ with speed $v$, resolve into $v_h = v\cos\theta$ and $v_v = v\sin\theta$, then apply the equations of motion to each component. The horizontal motion uses $d_h = v_h t$; the vertical motion uses $v = u + at$ and $s = ut + \tfrac{1}{2}at^2$ with $a = -g$. This independence is why a bullet fired horizontally and one dropped from the same height hit the ground at the same instant.

Newton's law of universal gravitation

Every mass attracts every other mass with a force proportional to the product of the masses and inversely proportional to the square of their separation. Because $G$ is tiny, the force between everyday objects is negligible; it becomes important only when at least one mass is astronomical, such as a planet or star. Doubling the separation quarters the force (the inverse-square dependence).

The gravitational field

This is why $g$ at the Earth's surface is about $9.8\ \text{N kg}^{-1}$ but decreases with height, and why the Moon (smaller mass) has a surface field of only about $1.6\ \text{N kg}^{-1}$. A field line diagram for a planet shows lines pointing radially inward, closer together near the surface where the field is stronger.

Examples in context

A long jumper or a shot put follows a parabolic path set by the independent horizontal and vertical motions; the optimum launch angle near $45^{\circ}$ balances range against height. Satellites and the International Space Station stay in orbit because the gravitational force $\frac{GMm}{r^2}$ provides exactly the centripetal force needed; astronauts feel weightless because they and the station fall together at the same $g$. Tides arise from the difference in the Moon's gravitational pull across the Earth. Spacecraft trajectories to other planets are computed from Newton's law of gravitation, using the inverse-square weakening of each body's field to plan slingshot manoeuvres.

Try this

Q1. State what happens to the horizontal velocity of a projectile during flight when air resistance is ignored. [1 mark]

• Cue. It stays constant (there is no horizontal force).

Q2. A stone is dropped from rest off a bridge and takes $2.0\ \text{s}$ to reach the water. Calculate the height of the bridge. Take $g = 9.8\ \text{m s}^{-2}$. [2 marks]

• Cue. $h = \tfrac{1}{2}gt^2 = \tfrac{1}{2} \times 9.8 \times 2.0^2 = 19.6\ \text{m}$.

Q3. State how the gravitational force between two masses changes if the distance between them is doubled. [1 mark]

• Cue. It becomes one quarter of its original value (inverse-square law).