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On this page you will find all formulas for the calculation of speed, acceleration, distance and time with or without initial speed. At the end of the page there is an example for a better understanding of these formulas, in which the acceleration, the final velocity and the average velocity are calculated.

Links to subpages:

• Calculator for distance, velocity, acceleration and time (Stopping Distance Calculator)
• Acceleration Calculator

The following formulas can be used to calculate the average speed v, the distance s or the time t required, where the average speed is constant.The first formula with the delta sign Δ represents the correct mathematical notation and is also called the difference quotient, because the difference between the distances is divided by the time difference:

Simplified notation of these formulas

The above formulas are known in a simplified notation, too. However, you have to keep in mind that distance and time are differences, see also the following examples on this page:

Meaning of the Variables

Attention to the units:
The units must always fit together! To convert a speed v given in km/h into m/s, simply divide the speed by 3.6:
v = 18 km/h --> 18/3.6 --> v = 5 m/s

If you multiply a speed v given in m/s by 3.6, you get the same speed in km/h:

v = 10 m/s --> 10⋅3.6 --> v = 36 km/h

Other possibility:
If you take distance in km and time in h, you get speed in km/h.

The average acceleration a, the change in velocity v or the required time t can be calculated by using the following formulas, whereby the average acceleration is constant:

Meaning of the Variables

The following formulas are based on the above definition of the average acceleration or its integral. The speed at time t₀ is called the initial velocity v₀ and the speed at time t₁ is called the final velocity v. The time difference Δt and the difference of distance Δs are represented in a simplified manner by t and s.

The following formulas apply to a uniform (= constant) acceleration or deceleration (= braking, negative acceleration) only with an initial velocity not equal to 0. Please note the hint on differences below the formulas!

Meaning of the Variables

Note on differences

• A negative value for the acceleration means that the vehicle is actually braking or decelerating.
• One must note that distance and time are actually differences. However, if s(t₀) and t₀ are equal to 0, this fact can be ignored. In principle, a starting distance is not included in the formulas, as it is not relevant for most tasks. However, it is possible to replace the distance s by the term s - s₀, as the following example shows.
• Below you find another example of calculating a time difference.

Example with starting distance s₀

If there is a starting distance, s is replaced by s - s₀ in the formula for the distance (1st line, 3rd column). Then move s₀ to the other side to get the distance s you are looking for:

This example shows that the above formulas can also be easily used in practice. You only need a stopwatch, which is available on every smartphone, and a measuring tape.

Specification

A garden train takes 6 seconds to reach the maximum speed by uniform acceleration, covering a distance of 9 m. Asumed that the initial speed is 0 m/s (acceleration from standstill)

• the time required
• the acceleration,
• the final velocity and
• the average speed has to be calculated.

Calculation of time

The time required for the acceleration process is the difference between the two times:
11 - 5 = 6 --> t = 6 s

Calculation of acceleration

If you insert in the formula of the 2nd line last column you get the acceleration you are looking for:
a = 2⋅s/t² = 2⋅9 m/(6 s)² --> a = 0.5 m/s²

Calculation of the final velocity

Now you can easily calculate the speed by simply choosing one of the three formulas out of the 3rd line. If you use the 2nd formula you get:
v = a⋅t = 0.5 m/s²⋅6 s --> v = 3 m/s
If you want to know the speed in km/h, multiply v by 3.6: 3⋅3.6 = 10.8 km/h

Calculation of the average speed

Since the distance is 0 at time t = 0 (= at the beginning) you only need to divide two numbers. Inserting in formula v = s/t results in:
v = 9 m / (6 s) --> v = 1.5 m/s = 1.5⋅3.6 km/h = 5.4 km/h

As you can see, the average speed is only half the final velocity.

If the acceleration is not constant, the use of the above formulas is not allowed. Instead, one calculates the acceleration, the velocity or the distance by differential or integral calculation.

You get the momentary acceleration a(t) by deriving the function of speed v(t) once or by deriving the function of distance s(t) twice after time t:

Distance s(t) is obtained by integrating the velocity v(t):

Page created on 31.03.2020. Last change: 28.11.2020.