URM Calculator
Uniform Rectilinear Motion: calculate distance, velocity or time with the formula d = v × t.
Uniform Rectilinear Motion (URM) is the simplest model of motion in physics: an object that moves in a straight line at constant velocity, with no acceleration. The relationship between distance, velocity and time is expressed by the formula d = v × t, which allows any of the three quantities to be calculated when the other two are known.
This calculator solves URM in all three possible variants: calculating the distance given velocity and time, calculating the velocity given distance and time, or calculating the time given distance and velocity. It also includes a position-time graph and supports different units (metres, kilometres, miles, km/h, m/s, mph, seconds, minutes, hours).
What is URM?
Uniform Rectilinear Motion (URM) is the motion in which an object moves in a straight line at constant velocity, i.e., without acceleration. The relationship between the quantities is:
Where d is the distance travelled, v is the velocity and t is the time elapsed.
How to solve URM problems step by step
The procedure is always the same. First, identify which quantity you are asked for and which ones you are given. Second, check the units: this is the most common mistake in exercises. If you mix km/h with seconds the result will be wrong, so convert everything into a consistent system (metres and seconds, or kilometres and hours). Remember that to go from km/h to m/s you divide by 3.6, and multiply by 3.6 for the reverse. Third, rearrange the formula d = v × t according to what you are looking for and substitute the values. Finally, check that the result makes physical sense: a distance or a time can never be negative.
Worked example
A train travels at a constant speed of 90 km/h for 2.5 hours. How far does it go? We apply d = v × t = 90 × 2.5 = 225 km. If instead we knew the distance (225 km) and the time (2.5 h), the velocity would be v = d / t = 225 / 2.5 = 90 km/h. And if we knew the distance and the velocity, the time would be t = d / v = 225 / 90 = 2.5 h. The key is that units must be consistent: if velocity is in km/h, time must be in hours for the distance to come out in kilometres.
Common reference speeds
| Situation | km/h | m/s |
|---|---|---|
| Person walking | 5 | 1,4 |
| Running (recreational) | 12 | 3,3 |
| Bicycle | 25 | 6,9 |
| Car in town | 50 | 13,9 |
| Car on motorway | 120 | 33,3 |
| Sound in air | 1.235 | 343 |
How to read the position-time graph
In URM, the graph of position against time is always a straight line, and its slope is precisely the velocity: the steeper the line, the faster the object moves. A horizontal line means the object is at rest. This contrasts with UARM, where the position-time graph is a parabola because the velocity changes continuously. If instead you plot velocity against time in URM you get a horizontal line, and the area enclosed under that line equals the distance travelled.