Uniform Motion Distance Calculator

Definition of uniform rectilinear motion

Uniform rectilinear motion is motion in which velocity is constant in both magnitude and direction. The distance s covered in time t satisfies s = v·t when the distance at time zero is zero. If the speed is not constant, the same product with the average speed v_avg = s/t gives the total distance. The calculator then assumes the number in the speed field is already that average.

Velocity is a vector. The distance on this page is the length of the path, not a signed displacement. Two routes with the same speed and the same time give the same distance even when they end in different places. Curves do not change the product as long as the speedometer reading stays constant.

Galileo, Newton, and the law of inertia

In Two New Sciences (Discorsi, 1638) Galileo separated uniform motion from accelerated motion. In uniform motion, equal times correspond to equal distances. That is the sentence form of s = v·t, written before calculus became the language of mechanics.

In the Principia (1687) Isaac Newton stated the first law: a body remains at rest or in uniform rectilinear motion unless a net force acts on it. Constant speed does not need a continuous push when the model has no resistance. Gravity in the solar system bends a probe’s path, so the motion is not strictly uniform and straight. A car on a highway holds a steady speed only while the engine balances drag.

Assumptions of the model s = v·t

The formula describes one speed for the whole time you enter. Acceleration in this model is zero. The time field is in hours whenever the speed is per hour. The fields do not take a direction, so the result is non-negative.

The calculator does not split a trip into segments, does not model traffic, and does not remove a stop you left inside the time. For a door-to-door result, the average must include the breaks. For driving only, leave the stopped time out. A negative speed or a negative time is rejected, because it does not describe distance in this model.

Derivation from constant velocity

Step 1. At constant velocity, v = Δs / Δt. Step 2. Therefore, Δs = v·Δt. Step 3. From a start with s(0) = 0, the distance is s = v·t. Step 4. In trip units, s_km = v_km/h · t_h, and the header converts that product to miles when US units are selected. Step 5. Conversions: v in m/s equals v in km/h divided by 3.6, and s in meters equals s in kilometers times 1,000. Minutes are t in hours times 60.

For 120 km/h and 2.5 h: s = 300 km = 300,000 m, v = 33.33 m/s, and t = 150 min. Pace, the reciprocal of speed in minutes per kilometer, is 60 / 120 = 0.5 min/km. The same arithmetic at 60 mph for 2 h gives 120 miles and a pace of 1 min/mi. At 10 km/h for 0.75 h the distance is 7.5 km and the pace is 6 min/km.

Average speed, section checks, and stops

The definition v_avg = s/t is a rearrangement of s = v·t. An average-speed check between two points knows the distance and the travel time, then computes v = s/t. Here you know v and t and compute s. The relation is the same. The unknown is different. A brief slowdown at one point does not change the average over the whole section.

A stop lowers the average speed if you include it in the time and do not add distance. One hour at 60 mph and one hour stopped gives an average of 30 mph over two hours and a distance of 60 miles. The calculator multiplies what you enter. It does not guess the breaks.

Units, scale consistency, and conversions

1 km/h = (1/3.6) m/s = (5/18) m/s. 1 h = 3,600 s = 60 min. The international mile is exactly 1.609344 km, so 120 km/h is about 74.56 mph and 300 km is 186.41 miles. BIPM defines the meter and the second. The mile and mph are conversions, not SI units. NIST SP 811 lists the factors. Time in the field stays in hours so that 45 is not multiplied as 45 hours.

A classroom round number for the speed of sound is 340 m/s. Six seconds from flash to thunder gives 340 × 6 = 2,040 m, or 2.04 km. That must not be computed as 340 × 6 kilometers. In these fields, 340 m/s is 1,224 km/h and 6 s is 6/3,600 h. The product is again 2.04 km. Light in vacuum travels 299,792,458 m in one second, about 300,000 km in classroom rounding. The mean Earth–Moon distance of about 384,400 km takes about 1.28 s. That is still s = v·t, on a scale different from an ordinary trip.

Sources

The definition, derivation, and units come from textbooks and metrology documents: D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, volume 1 (kinematics); H. D. Young and R. A. Freedman, University Physics, kinematics chapters; Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638; Isaac Newton, Philosophiæ Naturalis Principia Mathematica, 1687, Law I; BIPM, The International System of Units (SI), 9th edition; NIST Special Publication 811. This page is for educational purposes and does not replace a road survey or a navigation app.