Kinematics becomes much easier when a motion question is translated into a diagram, a list of known quantities, a suitable equation, and a units check. This guide explains a repeatable method for solving kinematics problems, then applies it to constant-velocity motion, uniformly accelerated motion, and projectile motion. Use the worked examples as a model for building your own step-by-step physics solutions rather than simply copying a final answer.
Overview
Kinematics describes motion without focusing on the forces that cause it. The main quantities are position or displacement, initial velocity, final velocity, acceleration, and time.
- Displacement, s: change in position, measured in metres (m). It includes direction.
- Distance: total path length travelled. It is a scalar and cannot be negative.
- Initial velocity, u: velocity at the start of the chosen time interval.
- Final velocity, v: velocity at the end of the interval.
- Acceleration, a: change in velocity per unit time, measured in metres per second squared (m/s²).
- Time, t: duration of the motion, measured in seconds (s).
For motion with constant acceleration, the standard motion equations are:
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
These equations are not interchangeable in every situation. Choose one that contains the quantity you want and the values you know. If acceleration changes during the motion, these constant-acceleration equations may not apply over the whole interval.
How to estimate
A reliable physics problem solver begins with the physical description, not with a formula. Follow this sequence:
- Define the system and interval. Decide what object is being studied and when the clock starts.
- Choose a positive direction. For example, take upward as positive or choose the direction of travel as positive.
- Draw a simple diagram. Mark the starting point, ending point, velocity direction, and any relevant height or angle.
- Write the known values with signs and units. A velocity to the left may be negative if right is positive. An acceleration opposing motion is also negative under that convention.
- Identify the unknown. It may be displacement, time, speed, velocity, or acceleration.
- Select an equation. Prefer the equation that uses the fewest unknown quantities.
- Substitute values and calculate. Keep units visible until the final line.
- Check the result. Ask whether the sign, size, units, and limiting behaviour make physical sense.
For a quick estimate, round only after setting up the equation. Early rounding can noticeably affect a result when squared terms such as t² appear. Also distinguish speed from velocity: speed gives magnitude only, while velocity includes direction.
Inputs and assumptions
Before calculating, record the assumptions that make the model valid. Typical introductory kinematics problems assume that the object can be treated as a particle, the motion occurs along a defined line or plane, and acceleration is constant during the selected interval.
For vertical motion near Earth’s surface, many courses use g = 9.8 m/s² as the magnitude of gravitational acceleration. If upward is positive, then the acceleration is a = −9.8 m/s². If downward is positive, it is a = +9.8 m/s². Either convention works if it is used consistently.
Convert inputs before substituting. For example, 72 km/h becomes 20 m/s because 72 ÷ 3.6 = 20. Angles in projectile problems should be interpreted consistently with the calculator setting, and horizontal and vertical components must use the same time interval.
A projectile model usually assumes air resistance is negligible. Under that assumption, horizontal acceleration is zero and vertical acceleration is −g when upward is positive. This is a model for calculation, not a complete description of every real flight.
Worked examples
Example 1: Constant velocity
A cyclist travels at a constant velocity of 6.0 m/s for 15 s. Find the displacement.
Because the velocity is constant, use:
s = vt
Substitute the inputs:
s = (6.0 m/s)(15 s) = 90 m
The displacement is 90 m in the direction of travel. The units reduce to metres because seconds cancel.
Example 2: Uniform acceleration from rest
A cart starts from rest and accelerates at 2.0 m/s² for 8.0 s. Find its final velocity and displacement.
Known values are u = 0 m/s, a = 2.0 m/s², and t = 8.0 s.
First find final velocity:
v = u + at = 0 + (2.0)(8.0) = 16 m/s
Now find displacement:
s = ut + ½at²
s = (0)(8.0) + ½(2.0)(8.0)² = 64 m
The cart finishes at 16 m/s and travels 64 m. A useful check is that the average velocity for constant acceleration is (0 + 16)/2 = 8 m/s, so the displacement is (8 m/s)(8 s) = 64 m.
Example 3: Vertical launch
A ball is thrown vertically upward at 14 m/s. Ignoring air resistance, estimate the time to reach its highest point and the maximum height above its launch point.
Take upward as positive. At the highest point, the final velocity is zero, so v = 0, u = 14 m/s, and a = −9.8 m/s².
Use v = u + at:
0 = 14 − 9.8t
t = 14 ÷ 9.8 ≈ 1.43 s
For height, use v² = u² + 2as:
0² = 14² + 2(−9.8)s
s = 196 ÷ 19.6 = 10 m
The ball rises approximately 10 m above its launch point. The negative acceleration does not mean the height is negative; it reflects the chosen upward-positive direction.
Example 4: Projectile components
A projectile is launched at 20 m/s at an angle of 30° above the horizontal. Resolve its initial velocity into components.
The horizontal and vertical components are:
ux = u cos θ = 20 cos 30° ≈ 17.3 m/s
uy = u sin θ = 20 sin 30° = 10.0 m/s
For an ideal projectile, use ax = 0 and ay = −9.8 m/s². The horizontal and vertical motions can then be solved separately using the same time. This component method is more dependable than trying to apply one one-dimensional equation to the entire angled motion.
When to recalculate
Recalculate whenever an input or assumption changes. In a homework problem, that may mean correcting a unit conversion, changing the sign convention, or using a different launch height. In an experiment, revisit the calculation when a measured time, distance, or angle is replaced by a new reading.
Also reconsider the model if the object’s acceleration is not constant, if air resistance is important, if the motion covers a large vertical distance, or if the object cannot reasonably be treated as a point mass. For repeated estimates, keep a small table containing each input, its unit, its sign, and the equation used. That makes recalculation quick and exposes transcription errors.
For revision, practise the same workflow with unfamiliar numbers: sketch the motion, list the variables, choose the direction, select the equation, calculate, and check. Use the Physics Problem Solver guide for a broader method-selection framework, and consult the GCSE physics equation list, A-Level revision notes, or AP Physics revision hub to match the notation and equation requirements of your course.
