First slide

Instantaneous acceleration

Question

The displacement x of a particle varies with time according to the relation $\large x = \frac{a} {b}(1 - e^{ - bt} )$ . Then

Moderate

A

At t = 1/b, the displacement of the particle in nearly (1/3) (a/b)
B

The velocity and acceleration of the particle at t = 0 are a and –ab respectively
C

Both 1 and 2
D

At t = 1/b, the displacement of the particle is (3/5) (a/b)

Solution

Velocity = $\frac{{dx}} {{dt}} = \frac{a} {b}[1 - e^{ - bt} ( - b)]$
$v = \frac{a} {b}[be^{ - bt} ];{\kern 1pt}$ At t = 0; v = a
Acceleration = $\frac{{dv}} {{dt}} = \frac{a} {b}[be^{ - bt} ( - b)]$ ; At t = 0; Acceleration = -ab

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