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Variable acceleration

Functions of time: Displacement, velocity, acceleration

Functions of time: Displacement, velocity, acceleration

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

Static particles

Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

Connected particles

Projectiles

Horizontal projection

Projectiles: Horizontal and vertical components

Projection at any angle

Equation of the trajectory of a projectile

Forces and friction

Resolving forces

Inclined planes

The coefficient of friction

Moments

Moments

Resultant moments

Equilibrium

Centre of mass

Tilting

Laminas

Forces and motion

Force diagrams

Forces as vectors

Forces and acceleration

Motion in two dimensions

Connected particles

Pulleys

Variable acceleration

Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

Displacement-time graphs

Velocity-time graphs

Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

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Hypothesis testing for zero correlation

Hypothesis testing I

Hypothesis testing

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One-tailed tests

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Binomial distribution

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The binomial distribution

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Calculating probabilities

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Probability: Tree diagrams

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Box plots

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Correlation

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Measures of central tendency: Mean, median, mode

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Variance and standard deviation

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Population and samples

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Vectors II

3D coordinates

Vectors in 3D

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Applications to mechanics: 3D vectors

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Numerical methods: Applications to modelling

Integration II

Integrating standard functions

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Differentiation II

Differentiating sin x and cos x

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Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

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Trigonometry and modelling

Addition formulae

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Solving trigonometric equations: Double angle formula

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Proving trigonometric identities

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Small angle approximations

Sequences and series

Sequences revision

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Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

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Functions

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

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Modulus inequalities

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Proof II

Proof by contradiction

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Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

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Integration I

Integrating x^n

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Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

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Gradients, tangents and normals

Increasing and decreasing functions

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Exponential functions

y = e^x

Exponential modelling

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Binomial expansion I

Pascal's triangle

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Binomial expansion

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Midpoints and perpendicular bisectors

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y = mx + c

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Proof I

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Proof by exhaustion

Explainer Video

Tutor: Mohammed

Summary

Functions of time: Displacement, velocity, acceleration

​​In a nutshell

When an object has variable acceleration it means the acceleration is changing over time. The displacement, velocity and acceleration can be modelled as functions of time.



Acceleration on a velocity-time graph

Maths; Variable acceleration; KS5 Year 12; Functions of time: Displacement, velocity, acceleration


A changing acceleration can be shown on a velocity-time graph. Acceleration is the rate of change of velocity, and can be found by working out the gradient of the curve. An increasing acceleration is represented by an increasing gradient, and a decreasing acceleration is represented by a decreasing gradient. 

Displacement, velocity and acceleration can be modelled as functions in terms of time. 



Displacement as a function of time

Displacement can be modelled as a function of time. These functions can be drawn out on a graph to help visualise the scenario. The displacement function can be used to work out the displacement at any given value for time, and also to identify times when there is no displacement - on a graph this is when the function crosses the xxx​-axis.


Example 1

A ball moves in a straight line. Its movement can be modelled by s=3t3−8ts=3t^3-8ts=3t3−8t, where t>0t>0t>0. Find:

a: The displacement of the ball at 444 seconds.

b: The time it takes for the ball to return to its origin.


a: At 444 seconds, t=4t=4t=4. Substitute into the formula for displacement:

s=3t3−8ts=3t^3-8ts=3t3−8t

s=3(43)−8(4)s=3(4^3)-8(4)s=3(43)−8(4)

​s=192−32=160 ms=192-32=160\ ms=192−32=160 m​​


The displacement of the ball at four seconds is 160 m‾\underline{160\ m}160 m​.


b: At the origin, s=0s=0s=0. Therefore:

3t3−8t=03t^3-8t=03t3−8t=0

Factorise:

​t(3t2−8)=0t(3t^2-8)=0t(3t2−8)=0

​​

For this to be true, either t=0t=0t=0 or 3t2−8=03t^2-8=03t2−8=0​. The time when the ball sets off is 000, therefore solve 3t2−8=03t^2-8=03t2−8=0:


3t2−8=03t^2-8=03t2−8=0​​

3t2=83t^2=83t2=8


t2=83t^2=\dfrac83t2=38​​​​


t=±83=±1.632993162t=\pm \sqrt{\dfrac83} = \pm 1.632993162t=±38​​=±1.632993162​​


The question says the model is only valid for t>0t>0t>0, therefore ignore the negative:​


The ball takes 1.64 s‾\underline{1.64\ s}1.64 s​ to return to its origin.



Velocity as a function of time

Velocity can also be modelled as a function of time. The function can be used to work out velocity at any given time, and to identify times when an object is stationary.


Example 2

The velocity of an object is given by v=2t2−26t+24v=2t^2-26t+24v=2t2−26t+24. Find the times when the object is at rest.


The object is at rest when v=0v=0v=0. Therefore:

2t2−26t+24=02t^2-26t+24=02t2−26t+24=0​​

Solve the quadratic:

​t2−13t+12=0t^2-13t+12=0t2−13t+12=0

(t−1)(t−12)=0(t-1)(t-12)=0(t−1)(t−12)=0

t=1, t=12t=1,\ t=12t=1, t=12​​​


The object is at rest at 1 s‾\underline{1\ s}1 s​ and 12 s‾\underline{12\ s}12 s​.​


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Speed, velocity and acceleration

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Speed, velocity and acceleration

Acceleration and velocity-time graphs

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Acceleration and velocity-time graphs

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Functions of time: Displacement, velocity, acceleration

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FAQs - Frequently Asked Questions

What is acceleration?

Acceleration is the rate of change of velocity.

What can I work out when the velocity if given as a function of time?

The velocity-time function can be used to work out velocity at any given time, and to identify times when an object is stationary.

What is variable acceleration?

Variable acceleration means an objects acceleration is changing over time.

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