URM Calculator

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.

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.

Two problems solved step by step

A 200-metre train travels at 72 km/h through a 1,300-metre tunnel. How long does it take to clear it completely? The typical mistake is using only the tunnel's length: the train has not cleared it until its last carriage passes the exit, so the distance travelled is 1,300 + 200 = 1,500 m. Converting the speed to metres per second, 72 / 3.6 = 20 m/s, the time is 1,500 / 20 = 75 seconds. Second problem: two cars leave at the same time from cities 300 km apart and drive towards each other at 90 and 110 km/h. Since they are approaching, their speeds add up: the closing speed is 200 km/h, and they meet after 300 / 200 = 1.5 hours. The first will have covered 135 km and the second 165 km, which add up to the original 300: checking that sum is the quick way to know whether the setup is right.

How to read a motion graph

On a position-versus-time graph, uniform rectilinear motion appears as a straight line, and its slope is exactly the velocity: the steeper it is, the faster the object moves. A horizontal line means the object is at rest, and a descending line means it is moving backwards, that is, its velocity is negative with respect to the direction chosen as positive. Two lines crossing mark the instant and position at which two objects meet, which is the graphical way of solving the two-car problem. On a velocity-versus-time graph, by contrast, uniform motion appears as a horizontal line, because velocity does not change, and the area under that line is the distance travelled. Knowing how to move from one graph to the other saves half the calculations in exams.