Using Calculus in 1D (AQA AS Maths): Revision Note
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Using Calculus in 1D
How is calculus used in kinematics?
s, v and a are all functions of time
Velocity, v, is the rate of change of displacement, s, with respect to time
Acceleration, a, is the rate of change of velocity, v, with respect to time
Differentiate to go from s to v and from v to a
Integrate to go from a to v and from v to s
There will be a constant of integration, c, each time you integrate
The use of calculus can be seen using velocity-time graphs
Acceleration is the gradient which is found using differentiation
Displacement is the area under the graph which is found using integration
How do I find c (the constant of integration) in kinematics?
The value of c can be found by substituting known values, as in integration for pure maths
Information to find c may be given in words – look out for common phrases …
“… starting from rest …”: this means v = 0 when t = 0
“… initially …” – this means t = 0
Definite integration in kinematics
Integrating acceleration between two values of time gives the change in velocity of the object between those times
Integrating velocity between two values of time gives the displacement of the object between those times
This could be different to the distance travelled between those times
To find the total distance travelled you would need to calculate the areas above and below the x-axis separately
Worked Example
The velocity, v m s-1, of a particle is modelled by v = t3 — 4t2 + 3t, where t is time in seconds.
(a) Find the acceleration of the particle in terms of 
.
(b) Find the displacement of the particle, from its starting position, after 3 seconds
Examiner Tips and Tricks
Look for key words and phrases that indicate scalar, rather than vector quantities:
“distance …”, “speed” and “magnitude of …” are scalar
Displacement, velocity, acceleration are vectors
A sketch of the velocity-time graph is always helpful as you can use it to see important features such as times when the object is stationary. This is especially useful you are asked to find the total distance travelled.
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