First slide

Position Path Length (Distance) And Displacement

Question

The position of a particle moving rectilinearly is given by $x = t^{3} - 3 t^{2} - 10$ . Find the distance travelled by the particle in the first 4 seconds starting from t = 0.

Difficult

A

24 units
B

36 units
C

15 units
D

50 units

Solution

$x = t^{3} - 3 t^{2} - 10$
$v = \frac{dx}{dt} = 3 t^{2} - 6 t$
Now, v = 0 gives
t = 0 and t = 2 sec.
Velocity will become zero at t = 2 sec., so particle will change direction after t = 2 sec.
At t = 0
$x_{(o \sec)} = - 10$
At t = 2 sec.
$x_{(2 \sec)} = 2^{3} - 3 {(2)}^{2} - 10 = 8 - 12 - 10 = - 14$
At t = 4 sec.
$x_{(4 \sec)} = 4^{3} - 3 {(4)}^{2} - 10$
= 64-48-10 = 6
Distance travelled = $x_{1} + x_{2}$
= $| - 14 - (- 10) | + | 6 - (- 14) | = 4 + 20 = 24$
Distance travelled = 24 units.

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