Piecewise Motion

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In general the motion of a particle is not smooth. Particles may change direction suddenly, collide, a string may become taut or new forces appear, thus changing the acceleration. In these circumstances, we can define for example, the velocity over certain intervals of time.

These points should be noted:

The displacement is always continuous. If the displacement is defined over several time intervals, the ends should ‘join’ up, so the particle never instantaneously disappears and reappears somewhere else.

Velocity can only change instantaneously in collisions, which may include situations where a string becomes taut, and acceleration can only change instantaneously when forces appear or disappear.

An insect moves in a straight line. The velocity is given by

when

i)Show that

ii)Findandand describe the motion for

iii)Sketchagainstfor

i)There are no collisions so the velocity is continuous atSubstitutinginto will give the same result as substitutingintohence

ii)Whenhence

Whenhence

(2)-(1) gives

Then from (1)

In the intervalthe insect decelerates from 12 m/s to -8m/s.

iii)The graph of v against t is drawn below.

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