Question

The velocity of a particle moving along a straight line increases according to the linear law v = $v_0$ + kx

where k is a constant . Then

1) the accelearation of the particle is k($v_0$ + kx)

2) the particle takes a time $\frac{1}{k}$ $lo{g}_e$ ($\frac{v_1}{v_0}$) to attain a velocity $v_1$

3) velocity varies linearly with displacement with slope of velocity displacement curve equal to k

4) data is insufficient to arrive at a conclusion.

• A. 1,2,3 are correct
• B. 2,3 are correct
• C. all of them are correct
• D. 1 and 2 are correct

Answer 1

The correct option is A

1,2,3 are correct

Options (1), (2), (3) are correct.

Accelearation = $\frac{dv}{dt}$ =kv

$\Rightarrow$ a = k ($v_0$ + kx)

Further

$\Rightarrow$ $\frac{dv}{dt}$ = kv

$\Rightarrow$ $\frac{dv}{v}$ = kdt

$\underset{v_0}{\overset{v_1}{\int }}$ $\frac{dv}{v}$ = k $\underset{0}{\overset{t}{\int }}$ ft

t = $\frac{1}{k}$ $lo{g}_e$ $\frac{v_1}{v_0}$

Since , v = $v_0$ + kx . Hence slope of velocity displacement curve is $\frac{dv}{dx}$ = k