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Variable acceleration

Constant acceleration formulae

Constant acceleration formulae

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Explainer Video

Tutor: Mohammed

Summary

Constant acceleration formulae

​​In a nutshell

When acceleration is constant, you can use the constant acceleration formula to find missing variables. You can use integration to derive these formulae.



Equations


DESCRIPTION

EQUATION

Velocity of an object with constant acceleration.
​v=u+atv=u+atv=u+at​​
Displacement of an object over a certain time.
​s=(u+v2)ts=\bigg(\dfrac{u+v}{2}\bigg)ts=(2u+v​)t​​
Velocity of an object with constant acceleration.
​v2=u2+2asv^2 = u^2+2asv2=u2+2as​​
Displacement of an object with constant acceleration
using initial velocity.
​s=ut+12at2s=ut+\dfrac{1}{2}at^2s=ut+21​at2​​
Displacement of an object with constant acceleration
using final velocity.
​s=vt−12at2s=vt-\dfrac{1}{2}at^2s=vt−21​at2​​

 

Variable definitions


QUANTITY NAME

SYMBOL

UNIT NAME

UNIT

DisplacementDisplacementDisplacement​​
​sss​​
​MetreMetreMetre​​
​mmm​​
​Initial velocityInitial\ velocityInitial velocity​​
​uuu​​
​Metres per secondMetres\ per\ secondMetres per second​​
​ms−1ms^{-1}ms−1​​
​Final velocityFinal\ velocityFinal velocity​​
​vvv​​
​Metres per secondMetres\ per\ secondMetres per second​​
​ms−1ms^{-1}ms−1​​
​AccelerationAccelerationAcceleration​​
​aaa​​
​Metres per second squaredMetres\ per\ second\ squaredMetres per second squared​​
​ms−2ms^{-2}ms−2​​
​TimeTimeTime​​
​ttt​​
​SecondsSecondsSeconds​​
​sss



Deriving the formula

Take a particle which is moving in a straight line with a constant acceleration of a ms−2a\ ms^{-2}a ms−2. The particle has an initial velocity of u ms−1u\ ms^{-1}u ms−1​ and an initial displacement of 0m0m0m. Find a formula for its velocity vvv and its displacement sss at time ttt.​


Velocity can be found by integrating acceleration:

​v=∫a dtv= \int{a}\ dtv=∫a dt

v=at+cv= at+cv=at+c​​​

The initial velocity is uuu. When t=0, v=ut=0,\ v=ut=0, v=u. Substitute:​

​u=a(0)+c=cu=a(0)+c =cu=a(0)+c=c​​

Therefore:

​v=u+at\boxed{v = u+at}v=u+at​​​


Displacement can be found by integrating velocity:

​s=∫v dts= \int{v}\ dts=∫v dt​​

Substitute the equation for velocity:

​s=∫u+at dts= \int{u+at}\ dts=∫u+at dt


​s=ut+12at2+cs= ut+\dfrac12at^2+cs=ut+21​at2+c

​​​

When t=0,s=0t=0, s=0t=0,s=0: ​

​0=u(0)+12a(02)+c0=u(0)+\dfrac12a(0^2)+c0=u(0)+21​a(02)+c


c=0c=0c=0​​​

Therefore:

​s=ut+12at2\boxed{s=ut+\dfrac12at^2}s=ut+21​at2​​​


Note: You may be asked to prove the constant acceleration formulae. You must be able to use integration to derive these formulae.


Example:

A cyclist moves in a straight line with a constant acceleration of 4 ms−24\ ms^{-2}4 ms−2. Given that his initial velocity is 10 ms−110\ ms^{-1}10 ms−1, show that his velocity at time ttt is given by v=10+4tv=10+4tv=10+4t.


Velocity is found by integrating acceleration:

v=∫a dtv=\int{a}\ dtv=∫a dt

v=at+cv=at+cv=at+c​​​

a=4a=4a=4:

v=4t+cv=4t+cv=4t+c​​

At t=0, v=10t=0,\ v=10t=0, v=10:

10=4(0)+c=c10=4(0)+c=c10=4(0)+c=c​​

​

Therefore, the cyclist's velocity at time ttt is given by v=10+4t‾\underline{v=10+4t}v=10+4t​.


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Acceleration and velocity-time graphs

Unit 1

Acceleration and velocity-time graphs

Constant acceleration: Deriving formulae

Unit 2

Constant acceleration: Deriving formulae

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Constant acceleration formulae

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FAQs - Frequently Asked Questions

How can you derive the formula for displacement?

Displacement can be found by integrating velocity.

How can you derive the formula for velocity?

Velocity can be found by integrating acceleration.

How can you derive the constant acceleration formulae?

You can use integration to derive these formulae.

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