The displacement of the particle varies with time according to the relation $\mathrm{x}=\frac{\mathrm{k}}{\mathrm{b}}[1-{\mathrm{e}}^{-\mathrm{bt}}]$. Then the velocity of the particle is:

(1) $k\left(e^{-bt}\right)$

(2) $\frac{k}{b^2{e}^{-bt}}$

(3) $kb{e}^{-bt}$

(4) None of these 

A particle is moving in a straight line and passes through a point O with a velocity of 6 ms^–1. The particle moves with a constant retardation of 2 ms^–2 for 4 s and there after moves with constant velocity. How long after leaving O does the particle return to O

(1) 3s

(2) 8s

(3) Never

(4) 4s

A bird flies for 4 s with a velocity of $|t-2|m/s$ in a straight line, where t is time in seconds. It covers a distance of

1. 2 m

2. 4 m

3. 6 m

4. 8 m

A body is projected vertically up with a velocity v and after some time it returns to the point from which it was projected. The average velocity and average speed of the body for the total time of flight are

(1) $\overrightarrow{v}/2$ and v/2

(2) 0 and v/2

(3) 0 and 0

(4) $\overrightarrow{v}/2$ and 0

Two cars P and Q start from a point at the same time in a straight line and their positions are represented by $x_p\left(t\right)=at+b{t}^2$ and $x_Q\left(t\right)=ft-t^2$. At what time do the cars have the same velocity?

(1) $\frac{a-f}{1+b}$

(2)$\frac{a+f}{2\left(b-1\right)}$

(3) $\frac{a+f}{2\left(1+b\right)}$

(4) $\frac{f-a}{2\left(1+b\right)}$

A particle moves a distance \(x\) in time \(t\) according to equation \(x = (t+5)^{-1}\). The acceleration of the particle is proportional to: 

A particle shows distance-time curve as given in this figure. The maximum instantaneous velocity of the particle is around the point:

         
1. B

2. C

3. D

4. A

A particle moves in a straight line, according to the law \(x=4a[t+a\sin(t/a)], \) where \(x\) is its position in metres, \(t\) is in seconds & \(a\) is some constant, then the velocity is zero at:

A point moves in a straight line so that its displacement is \(x\) m at time \(t\) sec, given by ${\mathrm{}}^{}$\(x^2= t^2+1\). Its acceleration in \(\text{m/s}^2\) at time \(1\) sec is:
1. \(\frac{1}{x}\)
2. \(\frac{1}{x} - \frac{1}{x^{3}}\)
3. \(\frac{2}{x}\)
4. \(-\frac{t^2}{x^3}\)

If the body covers one-third distance at speed v₁, next one third at speed v₂ and last one third at speed v₃, then average speed will be

(1) $\frac{v_1{v}_2+v_2{v}_3+v_3{v}_1}{v_1+v_2+v_3}$

(2) $\frac{v_1+v_2+v_3}{3}$

(3) $\frac{v_1{v}_2{v}_3}{v_1{v}_2+v_2{v}_3+v_3{v}_1}$

(4) $\frac{3v_1{v}_2{v}_3}{v_1{v}_2+v_2{v}_3+v_3{v}_1}$