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UARM Calculator

Uniformly Accelerated Rectilinear Motion: calculate final velocity and distance travelled.

Uniformly Accelerated Rectilinear Motion (UARM) describes the motion of an object moving in a straight line with constant acceleration. It is the model that describes free fall, vertical throws and the motion of vehicles that accelerate or brake uniformly. The UARM equations allow the velocity at any instant and the distance travelled to be calculated.

This calculator solves UARM given the initial velocity, acceleration and time values. It calculates the final velocity and distance covered, and generates a graph of velocity and position over time. The acceleration can be positive (the object accelerates) or negative (the object decelerates or moves in the opposite direction).

Motion data

Initial velocity v₀ (m/s)

Can be negative (opposite direction)

Acceleration a (m/s²)

Can be negative (deceleration)

Time t (s)

What is UARM?

Uniformly Accelerated Rectilinear Motion (UARM) is the motion in which an object moves in a straight line with constant acceleration. The velocity changes uniformly over time.

v(t) = v₀ + a · t

d(t) = v₀ · t + ½ · a · t²

v² = v₀² + 2 · a · d

The three UARM equations and when to use each

UARM is described by three equations derived from one another. The first, v = v₀ + a·t, relates velocity to time and is used when you want to know how fast the object is moving at a given instant. The second, d = v₀·t + ½·a·t², gives the distance travelled as a function of time and is the one that appears in most free-fall problems. The third, v² = v₀² + 2·a·d, is especially useful because it does not include time: it relates velocity and distance directly, which is very handy in braking problems where time is neither known nor asked for. Choosing the right equation is the step that saves the most time when solving an exercise.

Worked example

A car travels at 20 m/s and brakes with a constant acceleration of −4 m/s². How long does it take to stop and how far does it travel? For the time we use v = v₀ + a·t with v = 0: 0 = 20 + (−4)·t, giving t = 5 s. For the distance the third equation is better, as it does not depend on time: v² = v₀² + 2·a·d → 0 = 400 + 2·(−4)·d → d = 400/8 = 50 m. The car covers 50 metres before stopping, which explains why braking distance grows so quickly with speed: since it depends on v², doubling the speed quadruples the distance.

Gravitational acceleration on different bodies

Celestial body	Gravity (m/s²)
Moon	1,62
Mars	3,72
Venus	8,87
Earth	9,81
Jupiter	24,79
Sun	274

How to read UARM graphs

In UARM the graph of velocity against time is a straight line whose slope is the acceleration: rising if the object speeds up and falling if it brakes. The area under that line equals the distance travelled, a very useful trick for solving problems graphically. The position-time graph, by contrast, is a parabola: curvature upwards indicates positive acceleration and downwards, deceleration. If the parabola reaches a maximum and then descends, it means the object stopped and reversed direction, as happens in a vertical throw when the ball reaches its highest point.

Frequently asked questions

Free fall is the most characteristic example of UARM: an object falling solely under the action of gravity, with a constant acceleration of g ≈ 9.81 m/s² at the Earth's surface and zero initial velocity. A surprising consequence is that, ignoring air resistance, all bodies fall with the same acceleration regardless of their mass: a feather and a steel ball would hit the ground at the same time in a vacuum. When the object starts with an initial velocity upwards or downwards, the motion is called a vertical throw, but the equations applied are exactly the same.

Negative acceleration means the acceleration vector points in the opposite direction to the one taken as positive. If the object moves in the positive direction, a negative acceleration makes it slow down: this is the case of a car reducing speed. But avoid the automatic interpretation, because if the object is already moving in the negative direction, a negative acceleration speeds it up even more. The sign always depends on the reference frame you chose when setting up the problem, so it is worth defining it explicitly before you start calculating.

When the acceleration is zero (a = 0). Substituting a = 0 into the UARM equations, the first becomes v = v₀ (velocity does not change) and the second reduces to d = v₀ · t, which is exactly the URM formula. That is why URM can be understood as a particular case of UARM, the simplest one of all. In practice, any motion in which velocity stays constant over the stretch being studied can be treated as URM, even if there were accelerations before or after that stretch.

Air resistance is a force that opposes motion and grows with velocity, so the acceleration stops being constant and the UARM equations stop being exact. In a real fall, as the object gains speed the resistance increases until it matches the weight; at that moment the acceleration vanishes and the body falls at constant velocity, which is known as terminal velocity. A skydiver reaches about 200 km/h in a horizontal position before opening the parachute. Modelling these situations accurately requires differential equations, although for short falls and dense objects UARM remains a good approximation.

Because the distance depends on the square of the velocity, not on the velocity itself. From the equation v² = v₀² + 2·a·d, solving for braking distance gives d = v₀² / (2·|a|). This means doubling your speed multiplies the distance needed to stop by four, and tripling it multiplies it by nine. A car braking from 50 km/h needs about 15 metres, while from 100 km/h it needs around 60. On top of that you must add the distance covered during the driver's reaction time, which at 100 km/h amounts to almost 30 extra metres.

Average acceleration is the total change in velocity divided by the time elapsed: a = (v − v₀) / t. Instantaneous acceleration is the value the object has at a specific moment and corresponds to the derivative of velocity with respect to time. In UARM the two coincide, because the acceleration is constant by definition, and that is precisely why its equations are so simple. In motion with variable acceleration, such as a fall with air resistance, the distinction does matter and you need differential calculus.