Motion in one and two dimensions

What the exam asks

Expect a velocity-time or position-time graph and a question about a quantity the graph does not label. The commonest items are: find the displacement over an interval, identify where acceleration is greatest, describe what the body is doing at a named instant, and compare distance with displacement. The other half of the topic is a projectile with numbers chosen so the arithmetic is clean without a calculator. The trap that catches most candidates is treating the two graphs as interchangeable: a steep line means fast on a position-time graph and hard-accelerating on a velocity-time graph, and the same picture answers two different questions. Check the vertical axis label before reading anything off the line.

This topic covers how position, velocity and acceleration relate to each other, how to read that relationship off a graph, and what happens when a body moves in two dimensions at once. The equations of motion are supplied in the stem when they are needed, so nothing here is worth memorising. What is worth carrying in is the reading.

Almost every motion item is a graph item. A velocity-time graph is the workhorse: its gradient is acceleration and the area under it is displacement, and those two facts answer a whole family of questions without a single equation. A position-time graph is read differently, and confusing the two is the most expensive mistake in the topic.

Projectiles look like a new problem and are not. They are two one-dimensional problems running side by side, sharing nothing but a clock. Horizontal velocity is constant because nothing pushes horizontally; vertical velocity changes at g because gravity pulls down. The two never talk to each other, and once you believe that, most projectile questions collapse into one line of reasoning.

What to hold

• On a velocity-time graph the gradient is acceleration and the area between the line and the axis is displacement. Those two readings answer most of this topic.
• Area below the time axis on a velocity-time graph is negative displacement, so it subtracts from area above. Displacement is the signed total; distance is the sum of the magnitudes.
• On a position-time graph the gradient is velocity, not acceleration. Acceleration shows up as curvature.
• Where a velocity-time line crosses zero the body is momentarily at rest and about to reverse, and its displacement from the start is at a maximum or a minimum there.
• Negative acceleration does not mean slowing down. A body slows when acceleration opposes velocity, whatever signs they carry.
• A projectile's horizontal and vertical motions are independent: gravity has no horizontal component, so horizontal velocity never changes and vertical velocity changes at g throughout.
• Time of flight is set by the vertical problem alone. Horizontal speed cannot change how long a projectile stays up.
• Range is horizontal velocity multiplied by the time the vertical problem hands you, which is why doubling horizontal launch speed doubles the range.
• At the top of a projectile's arc the vertical velocity is zero but the acceleration is still g downwards, and the horizontal velocity is untouched.
• A launch at speed v and angle θ resolves into v cos θ horizontally and v sin θ vertically, and those two components are then two separate one-dimensional problems.
• Ignoring air resistance, a projectile's path is symmetric about its peak: it takes as long to come down as to go up, and the landing speed matches the launch speed at the same height.
• Speed is the magnitude of velocity, so a body can move at constant speed and still be accelerating if its direction changes.