Kinematics: Distance, Speed, Velocity and Acceleration

Distance is the total length of the path travelled. Displacement is the directed change from the starting position to the final position. Distance is a scalar: it has size but no direction. Displacement is a vector: it has both size and direction.

1.1 Follow the path, then compare the endpoints

Walk 6 m east, then 2 m west. Your distance is 6 + 2 = 8 m. Your displacement is 4 m east, because your final position lies east of your starting point. Choosing east as positive gives a displacement of +4 m. Choosing west as positive gives −4 m for the same journey.

The magnitude of displacement means its size without the direction. Distance is always greater than or equal to that magnitude. Equality holds when you travel along a straight line without reversing direction. Returning to the start gives zero displacement, not zero distance.

Average speed is total distance divided by total elapsed time. Average velocity is total displacement divided by total elapsed time. Their International System of Units (SI) unit is metres per second, written m/s. Instantaneous speed describes how fast an object moves at one moment. Instantaneous velocity also includes the direction at that moment.

2.1 A return journey

Suppose you walk 30 m east and return 30 m west in 60 seconds. Your average speed is 60 ÷ 60 = 1 m/s. Your average velocity is zero because your displacement is zero. Include any stops when calculating the average speed for the whole journey.

Constant velocity requires constant speed and constant direction. Constant speed alone does not guarantee constant velocity.

Average speed depends on the time spent moving, not just the two speed values. For equal distances at speeds v₁ and v₂, average speed is 2v₁v₂/(v₁ + v₂). The slower part takes longer, so it has a stronger effect on the overall average.

3.1 Work through an equal-distance journey

Travel 60 km at 30 km/h, then another 60 km at 60 km/h. The first part takes 2 hours, and the second takes 1 hour. Total distance is 120 km, while total time is 3 hours. Average speed is therefore 120 ÷ 3 = 40 km/h. The plain average, 45 km/h, gives the wrong answer for this journey.

For equal times at two speeds, the arithmetic mean does work. One hour at each speed gives 90 km in 2 hours. Average speed is then 45 km/h. When both speeds are equal, both averaging methods give that same speed.

Acceleration is the rate of change of velocity. Average acceleration is (final velocity − initial velocity) divided by elapsed time. Its SI unit is metres per second squared, written m/s². An object accelerates when its speed changes, its direction changes, or both change.

4.1 Read the sign with the direction

Choose east as positive. A velocity change from +10 m/s to +6 m/s in 2 seconds gives −2 m/s². The object slows because its acceleration points opposite to its velocity. A change from −10 m/s to −14 m/s gives the same acceleration, but the object speeds up. Negative acceleration therefore does not always mean slowing down.

4.2 Convert the bus example

A bus starts from rest and reaches 20 km/h in 10 minutes. Convert the time first: 10 minutes = 1/6 hour. Average acceleration is 20 ÷ (1/6) = 120 km/h². In SI units, 20 km/h is about 5.56 m/s, and 10 minutes is 600 seconds. Average acceleration is about 5.56 ÷ 600 = 0.00926 m/s². These changes determine the average acceleration; they do not prove the acceleration stayed constant.

Use these equations for straight-line motion with constant acceleration:

– v = u + at – s = ut + ½at² – v² = u² + 2as

Here, u is initial velocity and v is final velocity. Acceleration is a, elapsed time is t, and displacement is s. Use m/s for velocity, m/s² for acceleration, seconds for time, and metres for displacement. Choose one positive direction before substituting values. Velocity, acceleration and displacement must use that same sign convention.

5.1 A bicycle speeds up

A bicycle starts at 2 m/s and accelerates steadily at 1 m/s² for 4 seconds. Final velocity is 2 + (1 × 4) = 6 m/s. Displacement is (2 × 4) + (½ × 1 × 4²) = 16 m. Check with the third equation: 6² = 2² + (2 × 1 × 16).

Here, distance also equals 16 m because the bicycle never reverses direction. If an object reverses direction, s remains displacement. Calculate the path lengths separately when the question asks for distance. Do not use constant-acceleration equations across an interval with changing acceleration.

Slope measures the vertical change divided by the horizontal change. On a cumulative distance–time graph, slope gives speed. A steeper line means more distance covered each second. A horizontal line means the distance travelled is not increasing. A straight sloping line shows constant speed; a changing slope shows changing speed.

On a position–time graph, slope gives velocity instead. A downward slope can show motion in the negative direction. Cumulative distance cannot decrease, even when an object returns towards its starting point. The choice between distance and position changes what the graph means.

6.1 Read the units, not just the picture

An increase of 12 m in 3 seconds gives a slope of 4 m/s. A line's printed angle does not by itself give a physical speed or acceleration. Check the values and units represented by both axes.

The slope of a velocity–time graph gives acceleration. The velocity at time zero is the initial velocity. A horizontal line shows constant velocity and zero acceleration. The signed area between the graph and the time axis gives net displacement. Areas below the time axis count negatively. For distance, add the magnitudes of all the areas.

7.1 A journey in opposite directions

An object moves at +3 m/s for 2 seconds, then at −2 m/s for 2 seconds. The positive area is 6 m, and the negative area is −4 m. Net displacement is 6 − 4 = 2 m. Distance is 6 + 4 = 10 m. Equal positive and negative areas would give zero displacement, not zero distance.

For constant acceleration, v = u + at gives a straight velocity–time line. Its vertical intercept is u, and its slope is a. Do not confuse these two quantities.

Free fall is motion under gravity alone. Neglect air resistance near Earth's surface, and use approximately constant downward acceleration. The acceleration's magnitude is about 9.8 m/s². An object's mass does not change this ideal free-fall acceleration. Air resistance can make a feather fall differently from a coin in ordinary air.

8.1 Compare like starting conditions

Drop a coin and a feather together from the same height in a vacuum. If both start from rest, they reach the ground together. Equal acceleration alone does not guarantee equal arrival times. Different starting heights or initial velocities can change the result. The same comparison applies to two mangoes with different masses.

Choose upward as positive for a ball thrown straight upward. Its acceleration is approximately −9.8 m/s² throughout the flight. At the highest point, its velocity is momentarily zero. Gravity does not disappear at that point.

Rectilinear motion follows a straight path. A car on a straight road provides a simple example. A falling stone also follows a straight path when sideways motion is negligible.

Circular motion follows a circle around a centre. A point on a fan blade or clock hand follows this path. Uniform circular motion has constant speed, but its velocity keeps changing direction. Its acceleration points towards the centre. For speed v and radius r, the acceleration's magnitude is v²/r.

Periodic motion repeats after a fixed time interval. An ideal pendulum provides a familiar example. A vibrating guitar string approximately repeats its motion, but damping gradually reduces its vibration. Not every motion repeats, and not every repeating motion follows a circle.

9.1 Why a rolling ball needs two descriptions

A rolling ball translates as its centre moves forward. It also rotates around its centre. A marked point on the ball circles the moving centre. Relative to the ground, that point does not generally follow a fixed circle. Always identify the reference point before naming the path.

A projectile moves under gravity after launch when other forces are neglected. In the usual near-surface model, ignore air resistance and treat gravity as constant. Horizontal velocity stays constant because horizontal acceleration is zero. Vertical velocity changes because gravity acts downward.

10.1 Follow a ball over its curved path

Imagine a ball launched diagonally upward. The horizontal motion continues while the vertical motion slows. At the highest point, vertical velocity is zero. Horizontal velocity remains nonzero for an oblique launch. The ball therefore need not stop at the top. Its downward acceleration remains approximately 9.8 m/s².

After the highest point, vertical velocity points downward. The curved path combines horizontal motion with vertical acceleration. Do not treat the whole velocity as zero merely because its vertical part becomes zero.