Motion with Constant Acceleration

Kinematic model describing straight-line motion with constant acceleration and uniform change in velocity.

Conceptual context

Uniformly accelerated rectilinear motion (UARM) describes a body that moves along a straight path with a constant acceleration. As a consequence, the velocity increases or decreases uniformly over time: it changes by equal amounts during equal intervals of time.

It is one of the most useful models in kinematics. Almost every everyday motion involves acceleration, from a car pulling away to an object falling under the gravitational acceleration. Galileo showed that, without air resistance, bodies fall with the same acceleration regardless of their mass, which made it possible to describe the change in velocity precisely.

🟢 Essential level

Imagine a 100-metre race. When the gun fires, your velocity is zero, but every second you go faster. That steady increase is UARM: the rate at which the velocity changes is always the same.

The key is constant acceleration. If your velocity increases by 2 m/s every second, that rate stays the same throughout the run. To describe the initial state of the moving body we need three data: the initial position, the initial velocity (the starting speed) and the acceleration.

Unlike URM, the distances covered in equal times are not equal here. As you go faster and faster, you cover more distance in each second than in the one before: the distance covered grows ever faster with time while the body keeps speeding up.

[!TIP] Acceleration is the quantity that defines the change. If it has the same sign as the velocity, the object speeds up. If they have opposite signs, the object loses speed.

🔵 Formal level

Basic formulas

In UARM the acceleration is constant. Therefore the velocity changes linearly with time and the position changes quadratically. The analysis requires integrating these variations into a consistent system of equations that relates space, time and velocity.

The fundamental equations. We have four equations to describe the motion at any instant, starting from the initial position and the initial velocity:

1. Velocity equation: The final velocity is a linear function of the time, and the slope of that function is the acceleration:

Velocity-time equation

2. Position-time equation: It is a quadratic function: the term in time squared makes the graph of position against time a parabola.

Position-time quadratic equation

Real student questions

• Does a negative acceleration mean braking? Not necessarily. It brakes only if the velocity is positive. If both are negative, the object gains speed in the negative direction.
• Why does time appear squared? If the body starts from rest, the displacement is the area of a triangle on the velocity-against-time graph, whose base and height both grow with time. That is why the displacement grows with the square of the time.
• Can the velocity be zero while there is acceleration? Yes. At the highest point of a vertical throw the velocity is zero, and gravitational acceleration is still acting.

Final synthesis

UARM describes motion with constant acceleration: velocity varies linearly with time and position is a quadratic function of time. Its four equations solve one-dimensional kinematics problems whenever the acceleration is constant over the interval studied.