Cambridge Additional Mathematics

MOTION

DEMO

Be careful not to

confuse speed

with displacement.

St

st

()

()

384 Applications of differential calculus (Chapter 14)

ZEROS:

Phrase used in a question t s v a

initial conditions 0

at the origin 0

stationary 0

reverses 0

maximum or minimum displacement 0

constant velocity 0

maximum or minimum velocity 0

When a particle reverses

direction, its velocity must

change sign.

This corresponds to a local

maximum or local minimum

distance from the origin O.

SPEED

As we have seen, velocities have size (magnitude) and sign (direction). In contrast, speed simply measures

how fastsomething is travelling, regardless of the direction of travel. Speed is ascalarquantity which has

size but no sign. Speed cannot be negative.

Thespeedat any instant is the magnitude of the object’s velocity.

If S(t) represents speed then S=jvj.

To determine when the speed S(t) of an object P with displacements(t) is

increasing or decreasing, we use asign test.

² If the signs of v(t) and a(t) are the same (both positive or both

negative), then the speed of P is increasing.

² If the signs of v(t) and a(t) are opposite, then the speed of P is

decreasing.

Discovery Displacement, velocity, and acceleration graphs

In this Discovery we examine the motion of a projectile which is fired in a vertical

direction. The projectile is affected by gravity, which is responsible for the projectile’s

constant acceleration.

We then extend the Discovery to consider other cases of motion in a straight line.

What to do:

1 Click on the icon to examine vertical projectile motion.

Observe first the displacement along the line, then look at the velocity which is the rate of change

in displacement. When is the velocity positive and when is it negative?

2 Examine the following graphs and comment on their shapes:

² displacementvtime ² velocityvtime ² accelerationvtime

3 Pick from the menu or construct functions of your own choosing to investigate the relationship

between displacement, velocity, and acceleration.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\384CamAdd_14.cdr Monday, 7 April 2014 10:33:08 AM BRIAN