The kinematic equations describe motion in a straight line when acceleration is constant. They connect position, displacement, velocity, acceleration, and time, which makes them a core tool for solving many physics problems. They are often called the SUVAT equations because they use displacement s, initial velocity u, final velocity v, acceleration a, and time t.
These equations matter because they let you predict motion without needing to analyze every instant separately.
Each equation is useful when one variable is missing from the problem. For example, v = u + at is best when displacement is not needed, while v^2 = u^2 + 2as is best when time is not given. The equations apply to objects such as cars speeding up, balls falling vertically, and carts moving along tracks, as long as acceleration stays constant.
Choosing a positive direction and keeping signs consistent are just as important as substituting numbers.
Understanding Physics: The Kinematic Equations
A useful way to understand these equations is through graphs. On a velocity against time graph, constant acceleration makes a straight sloping line. The gradient of that line is the acceleration.
A steeper slope means velocity changes more quickly each second. The area under the line gives displacement. This area can be split into a rectangle from the initial velocity and a triangle caused by acceleration.
Adding those areas produces the displacement equation with initial velocity, acceleration, and time. The average velocity equation comes from treating the area as a trapezium.
These are not separate rules to memorise without meaning. They are different descriptions of the same graph.
Signs carry physical meaning, so choose a positive direction before using any values. If upward is positive, a ball thrown upward has positive initial velocity. Its acceleration due to gravity is negative because gravity points down.
As the ball rises, its velocity falls to zero at the highest point. It then becomes negative as the ball moves down. Its displacement can be zero when it returns to its launch height, even though it has travelled a considerable distance.
Distance never has a negative value, but displacement can. Confusing these two ideas is a common source of wrong answers.
The word constant is the condition that makes the equations reliable. Near Earth’s surface, gravity gives falling objects an acceleration of about nine point eight metres per second squared downward, provided air resistance is small. A dropped ball, a stone thrown vertically, and a trolley pulled with a steady force can often be modelled this way.
A car in normal traffic usually cannot. Its driver brakes, changes gear, turns corners, and meets changing resistance. For a changing acceleration, a velocity against time graph becomes curved.
One set of constant acceleration equations cannot describe the whole journey. It may still work for a short section where the acceleration is close to steady.
In calculations, begin by writing down what each known value represents. State the chosen positive direction, then attach signs to velocities, accelerations, and displacements. Convert units before substituting values.
Kilometres per hour must be changed to metres per second if the rest of the data uses standard units. Check the answer against the situation. A braking vehicle should have a final speed lower than its initial speed.
An object thrown upward should slow down before reversing direction. Squared velocity equations can produce two mathematical roots in some problems.
Select the root that fits the stated direction and moment of motion. Clear diagrams, labelled values, and a quick graph sketch prevent many mistakes before any arithmetic begins.
Key Facts
- v = u + at relates final velocity, initial velocity, acceleration, and time.
- s = ut + 1/2 at^2 gives displacement when initial velocity, acceleration, and time are known.
- s = 1/2(u + v)t gives displacement using average velocity when acceleration is constant.
- v^2 = u^2 + 2as relates velocities, acceleration, and displacement without using time.
- For constant acceleration, average velocity is v_avg = (u + v)/2.
- Use consistent units: displacement in meters, time in seconds, velocity in m/s, and acceleration in m/s^2.
Vocabulary
- Displacement
- Displacement is the change in position of an object, including direction.
- Initial velocity
- Initial velocity is the velocity of an object at the start of the time interval being studied.
- Final velocity
- Final velocity is the velocity of an object at the end of the time interval being studied.
- Acceleration
- Acceleration is the rate at which velocity changes with time.
- Constant acceleration
- Constant acceleration means the velocity changes by equal amounts during equal time intervals.
Common Mistakes to Avoid
- Using the equations when acceleration is not constant is wrong because SUVAT equations assume a single constant value of a throughout the motion.
- Mixing up distance and displacement is wrong because displacement includes direction and can be positive or negative depending on the chosen axis.
- Forgetting the sign of acceleration is wrong because slowing down or motion in the negative direction may require a negative value for a or v.
- Choosing an equation before identifying known and unknown variables is wrong because the best equation is the one that avoids the missing variable.
Practice Questions
- 1 A car starts from rest and accelerates at 3.0 m/s^2 for 8.0 s. Find its final velocity and displacement.
- 2 A ball is thrown straight upward with an initial velocity of 20 m/s. Using a = -9.8 m/s^2, find its velocity after 1.5 s and its displacement from the release point.
- 3 A cyclist moves forward while slowing down at a constant rate. Explain how the signs of velocity and acceleration should be chosen if forward is the positive direction.