Cambridge Additional Mathematics

Applications of differential calculus (Chapter 14) 381

VELOCITY

Theaverage velocityof an object moving in a straight line in the time interval from t=t 1 to t=t 2

is the ratio of the change in displacement to the time taken.

If s(t) is the displacement function then average velocity=

s(t 2 )¡s(t 1 )

t 2 ¡t 1

.

On a graph of s(t) againsttfor the time interval from t=t 1 to t=t 2 , the average velocity is the

gradient of a chord through the points (t 1 ,s(t 1 )) and (t 2 ,s(t 2 )).

InChapter 13we established that the instantaneous rate of change of a quantity is given by its derivative.

If s(t) is the displacement function of an object moving in a straight line, then

v(t)=s^0 (t) = lim

h! 0

s(t+h)¡s(t)

h

is theinstantaneous velocityor

velocity functionof the object at timet.

On a graph of s(t) againstt, the instantaneous velocity at a particular time is the gradient of the tangent

to the graph at that point.

ACCELERATION

If an object moves in a straight line with velocity function v(t) then:

² theaverage accelerationfor the time interval from t=t 1 to t=t 2 is the ratio of the change in

velocity to the time taken

average acceleration=

v(t 2 )¡v(t 1 )

t 2 ¡t 1

² theinstantaneous accelerationat timetis a(t)=v^0 (t) = lim

h! 0

v(t+h)¡v(t)

h

.

UNITS

Each time we differentiate with respect to timet, we calculate a rate per unit of time. So, for a displacement

in metres and time in seconds:

² the units of velocity are m s¡^1 ² the units of acceleration are m s¡^2.

Discussion

#endboxedheading

² What is the relationship between the displacement function s(t) and the acceleration function

a(t)?

² How are the units of velocity and acceleration related to their formulae? You may wish to research

“dimensional analysis”.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\381CamAdd_14.cdr Monday, 7 April 2014 10:24:23 AM BRIAN