Formulas for Speed, Acceleration, Distance & Time

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.

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Formulas for Average Velocity, Distance and Time

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:

calculation-constant-average-speed by the difference quotient
Calculation of the (constant) average speed by the difference quotient
Formulas for distance and time

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:

  • Example with starting distance s0 ≠ 0
  • Example with starting time t0 ≠ 0

 

This fact can be ignored if the starting distance s0 and the starting time t0 are equal to 0.

 

Formulas for speed, distance and time at constant speed
Formulas for speed, distance and time at constant speed

 

Meaning of the Variables

v

average velocity in m/s in the interval [t0; t1]

('velocity', therefore the abbreviation v)

s or Δs

distance or route in m in the interval [t0; t1];

sometimes d is used instead of s

t or Δt time required in s ('time', therefore the abbreviation t)
s(t1) distance at time t1
s(t0) distance at time t0 (this value is usually 0)

 

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.

Formulas for Average Acceleration

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:

Calculation of the (constant) average acceleration by the difference quotient
Calculation of the (constant) average acceleration by the difference quotient
Formulas for velocity and time

 

Meaning of the Variables

a

average acceleration in m/s² in the interval [t0; t1]

('velocity', therefore the abbreviation v)

Δv

Change of velocity (= velocity difference) in m/s in the interval [t0; t1];

Δt time required in s
v(t1) or v1 velocity at time t1 (final velocity)
s(t0) or v0 velocity at time t0 (starting or initial velocity); this value is usually 0

 

 

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.

Formuals for Uniform Acceleration - Starting Velocity ≠ 0

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!

 

Formulas for braking distance, acceleration, (starting) velocity and time-starting velocity is not 0-condition-constant acceleration
Formulas for calculating braking distance, acceleration, (starting) velocity and time while starting velocity is not 0; condition: constant acceleration.

 

Meaning of the Variables

v final velocity in m/s
v0  starting velocity in m/s
a

acceleration or deceleration in m/s²

s (braking) distance in m
t time needed in s

 

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:

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Formulas for Uniform Acceleration - without Starting Velocity

These formulas apply to constant acceleration and deceleration, whereby both the initial velocity and the initial distance must be zero.

In principle, these are the same formulas as above, but the initial speed v0 is set to zero. The 5th line in the formula collection above is completely omitted.

 

Please note the hint on differences in the previous chapter!

 

Formulas for initial velocity equal to 0-constant acceleration
Formulas for an initial velocity equal to 0; condition: constant acceleration.

Simple Example

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.

Current Velocity & Momentary Acceleration

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.

Main Formulas

The current velocity v(t) at any time t is calculated by deriving the function of distance s(t) once from time t (= differential quotient):

Current Speed by differentiating
Current Speed by differentiating

If the momentary acceleration is known, the function of acceleration a(t) must be integrated after time t in order to get the current velocity v(t):

Current Speed by integrating
Current Speed by integrating

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:

Momentary Acceleration
Momentary Acceleration
Momentary Acceleration
Momentary Acceleration

 

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

Distance
Distance

Meaning of the Variables

v(t)

Function of current speed

s(t)

Function of distance

a(t)

Function of momentary acceleration

t time

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Page created on 31.03.2020. Last change: 28.11.2020.