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Acceleration and velocity-time graphs

Acceleration and velocity-time graphs

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Summary

Acceleration and velocity-time graphs

In a nutshell

Acceleration is a vector, and is the rate of change of velocity. Acceleration over a period of time can be calculated by determining the gradient of a velocity time graph.


Equations

DESCRIPTION 

EQUATION

Velocity
​v=ΔsΔtv = \dfrac{\Delta s}{\Delta t}​​
Acceleration
​a=ΔvΔta = \dfrac{\Delta v}{\Delta t}​​


Variables

QUANTITY NAME

SYMBOL

DERIVED UNIT

SI UNIT

​speedspeed​​
​vv
​ms−1ms^{-1}​​
​ms−1ms^{-1}​​
​velocityvelocity​​
​vv​​
​ms−1ms^{-1}​​
​ms−1ms^{-1}​​
​displacementdisplacement​​
​ss​​
​mm​​
​mm​​
​timetime​​
​tt​​
​ss​​
​ss​​
​accelerationacceleration​​
​aa​​
​ms−2ms^{-2}​​
​ms−2ms^{-2}​​



Acceleration

Acceleration is defined as the rate of change of velocity. It is a vector, as it has both a magnitude and a direction. A negative acceleration is also sometimes referred to as a deceleration. It can be calculated using the following equation:


​a=ΔvΔta = \dfrac{\Delta v}{\Delta t}


Note: due to this definition, an object can accelerate by changing its direction or speed, or both.


Velocity-time graphs

The motion of an object can also be expressed on a velocity-time graph. On this graph, the y-coordinate represents the instantaneous velocity of the object. The gradient of the graph represents the acceleration. If it is a straight line acceleration is constant, if it is curved the acceleration is changing. 


The area under the graph is the displacement travelled. For a straight-line graph, this is easy to calculate, as the area under the graph is a series of triangles and rectangles which you can just add the area of together. 

 

Physics; Motion; KS5 Year 12; Acceleration and velocity-time graphs
1. 
In this section, the shape is that of a rectangle, so you would need to multiply the base and the height of the shape to find the area.
2.
In this section, the shape is that of a triangle, so you would need to multiply the base and the height together and divide by 2.


Example:

The velocity-time graph of a smart car is shown in the figure below. Calculate the total distance travelled by the car in the 50 s50\ s period.

Physics; Motion; KS5 Year 12; Acceleration and velocity-time graphs


First state the equation:

​distance travelled=area under the graphdistance \ travelled= area \ under \ the \ graph


Then calculate the answer:

​0<t<10 s:0.5×60×10=300 m10<t<40 s:60×30=1,800 m40<t<50 s:0.5×60×10=300 marea under=300+1,800+300=2,400 m0<t<10 \ s: 0.5 \times 60 \times 10 = 300\ m \\[0.05in] \newline 10<t<40 \ s: 60 \times 30 = 1,800 \ m \\[0.05in] \newline 40<t<50 \ s:0.5 \times 60 \times 10 = 300\ m\newline \\[0.1in] area \ under = 300 + 1,800 + 300\newline = 2,400\ m

​

The distance travelled by the smart car in the 50 s50 \ s period is 2400 m‾\underline{2400\ m}.​


Calculating displacement for changing accelerations

As described before, when the acceleration is changing, the velocity-time graph will show a curved line. This means that the area under the curve has to be estimated by drawing a series of shapes with calculatable areas (triangles, rectangles and trapeziums). The total area will be the sum of the individual shapes areas. 


Physics; Motion; KS5 Year 12; Acceleration and velocity-time graphs


To get the instantaneous velocity, draw a tangent to the curve at the specified time and calculate the gradient of this tangent. 



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Learn with Basics

Length:
Motion and speed

Unit 1

Motion and speed

Speed, velocity and acceleration

Unit 2

Speed, velocity and acceleration

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Acceleration and velocity-time graphs

Unit 3

Acceleration and velocity-time graphs

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FAQs - Frequently Asked Questions

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