A.1 Kinematics

Choose a reference

Position $\vec r$ specifies where an object is relative to a chosen origin. A position is not meaningful without a reference frame and coordinate direction.

Track change in position

Velocity $\vec v$ describes how position changes with time. It is a vector: its direction is the direction of motion at that instant, and on a curved path the velocity arrow is tangent to the path.

Track change in velocity

Acceleration $\vec a$ describes how velocity changes with time. A change in speed, direction, or both is acceleration; an object can accelerate even while its instantaneous speed is zero.

Common trap

Do not use “velocity” as a synonym for speed. Speed gives only magnitude; velocity also requires direction relative to the chosen coordinate system.

Velocity is a rate

Velocity is the rate of change of position:

$$\vec v=\frac{d\vec r}{dt}$$

Over a finite interval, average velocity is displacement divided by elapsed time.

Acceleration changes velocity

Acceleration is the rate of change of velocity:

$$\vec a=\frac{d\vec v}{dt}$$

A constant acceleration gives equal changes in velocity during equal time intervals.

Read the gradient

On a position–time graph, the gradient represents velocity. On a velocity–time graph, the gradient represents acceleration. The graph’s slope, not its height alone, carries the rate-of-change meaning.

Common trap

Distance divided by time gives average speed, not instantaneous velocity. Likewise, a large velocity does not imply a large acceleration unless the velocity is changing rapidly.

Displacement is a vector

Displacement is the change in position:

$$\Delta\vec r=\vec r_{final}-\vec r_{initial}$$

It has a magnitude and a direction from the initial position to the final position.

Ignore the route for displacement

The path taken between the two positions does not determine displacement. A curved or complicated journey can still have a straight-line displacement between its endpoints.

Use components when needed

For perpendicular changes, resolve displacement into components and combine them vectorially. A signed one-dimensional displacement is positive or negative according to the chosen axis.

Common trap

A return to the starting point gives zero displacement even though the distance travelled is non-zero.

Distance

Distance is the total path length travelled. It is a scalar, so it has magnitude only and cannot be negative.

Displacement

Displacement is the vector change in position from start to finish. Its magnitude is the shortest endpoint-to-endpoint separation, not generally the length of the route.

Match the average quantity

Average speed uses total distance divided by total time. Average velocity uses displacement divided by total time. A route with turns can therefore have average speed greater than the magnitude of average velocity.

Common trap

For a complete oscillation, the displacement is zero but the distance is four times the amplitude. Choose the quantity named in the question before substituting.

Average values use an interval

Average speed is total distance divided by total time. Average velocity is displacement divided by elapsed time. Average acceleration is change in velocity divided by elapsed time.

Instantaneous values use one moment

An instantaneous value describes the motion at a particular instant. On a position–time graph, instantaneous velocity is the tangent gradient at that instant; on a velocity–time graph, instantaneous acceleration is the tangent gradient.

Connect graph quantities

The gradient of a velocity–time graph is acceleration, and the area under it is displacement. A constant acceleration therefore produces a straight-line velocity–time graph.

Common trap

Do not use the average gradient when a question asks for an instantaneous value. Use the tangent at the specified time.

Uniform-acceleration model

SUVAT equations apply when acceleration is constant along the chosen one-dimensional axis:

$$v=u+at,\quad s=ut+\frac12at^2,\quad v^2=u^2+2as,\quad s=\frac{u+v}{2}t$$

Choose an equation

List the known and unknown quantities $s,u,v,a,t$. Select an equation containing the required unknown and only known quantities; keep signs consistent with the positive direction.

Check the model

A constant acceleration means equal changes in velocity in equal time intervals. Use separate horizontal and vertical equations only when the motion has been resolved into independent components.

Common trap

Do not use SUVAT when acceleration varies significantly with time or position. A formula can produce a neat number while still violating the model’s constant-acceleration assumption.

Uniform acceleration

Acceleration is uniform when the velocity changes by equal amounts in equal time intervals. The velocity–time graph is a straight line with constant gradient.

Non-uniform acceleration

Acceleration is non-uniform when its magnitude or direction changes. The velocity–time graph then has a changing gradient, and a single SUVAT value cannot describe the entire interval.

Model versus reality

A constant-acceleration model can be useful over a limited interval even when real forces vary. State the approximation and identify the neglected force or changing condition.

Common trap

A curved trajectory does not by itself prove that acceleration is non-uniform: projectile motion without drag has constant downward acceleration while its velocity direction changes.

Separate the axes

With negligible fluid resistance, projectile motion is independent horizontal and vertical motion. Resolve the launch velocity:

$$u_x=u\cos\theta,\qquad u_y=u\sin\theta$$

Horizontal motion

There is no horizontal acceleration in the ideal model, so $v_x=u_x$ and $x=u_xt$. Use the horizontal displacement to find time or horizontal speed.

Vertical motion

Use one-dimensional constant-acceleration equations vertically, usually with $a_y=-g$ if upward is positive. The horizontal and vertical equations share the same time $t$.

Common trap

Do not use the launch speed as the horizontal speed. Resolve it first, and do not use the horizontal time independently of the vertical motion.

Drag opposes motion

Fluid resistance acts opposite the instantaneous velocity. Its magnitude depends on speed, shape, fluid and orientation, so the net force and acceleration can change throughout the flight.

Compare with the ideal path

Compared with no-drag motion, drag reduces horizontal speed, shortens range and usually lowers the maximum height. The trajectory is not the symmetric parabola predicted by the ideal model.

Terminal speed

During a vertical fall, drag grows as speed grows. When drag balances weight, the net force and acceleration become zero and the object continues at terminal speed.

Common trap

At the top of a drag-affected trajectory, the velocity may be momentarily horizontal or zero, but the acceleration is determined by the resultant forces, not by the velocity alone.