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 of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x)=$\beta x^{-2n}$ where, $\beta$ and n are constants and x is the position of the particle. The acceleration of the particle as a function of x, is given by

1. $-2n{\beta }^2{x}^{-2n-1}$

2. $-2n{\beta }^2{x}^{-4n-1}$

3. $-2{\beta }^2{x}^{-2n+1}$

4. $2n{\beta }^2{e}^{-4n}$⁺¹

A stone falls under gravity. It covers distances h₁, h₂ and h₃ in the first 5 seconds, the next 5 seconds and the next 5 seconds respectively. The relation between h₁, h_2, and h₃ is

1. h₁=2h₂=3h₃

2. h₁=h₂/3=h₃/5

3. h₂=3h₁ and h3=3h₂

4. h₁=h₂=h₃

The motion of a particle along a straight line is described by equation
         \(x=8+12 t-t^3\)

where \(x\) is in metre and t is in second. The retardation of the particle when its velocity becomes zero is:

A boy standing at the top of a tower of 20 m height drops a stone. Assuming g=$10 {\mathrm{ms}}^{-2}$, the velocity with which it hits the ground will be:
1. 20 m/s                                         
2. 40 m/s
3. 5 m/s                                           
4. 10 m/s

A particle covers half of its total distance with speed ${\mathrm{v}}_1$ and the rest half distance with speed ${\mathrm{v}}_2.$ Its average speed during the complete journey is 

(1) $\frac{{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$                                         

(2) $\frac{2{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$

(3) $\frac{2{\mathrm{v}}_1^2{\mathrm{v}}_2^2}{{\mathrm{v}}_1^2+{\mathrm{v}}_2^2}$                                         

(4)  $\frac{{\mathrm{v}}_1+{\mathrm{v}}_2}{2}$

A ball is dropped from a high rise platform at t=0 starting from rest. After 6s another ball is thrown downwards from the same platform with a speed v. The two balls meet at t=18 s. What is the value of v? (take g=10 $m{s}^{-2}$)

1. $75 m{s}^{-1}$                                   2. $55 m{s}^{-1}$

3. $40 m{s}^{-1}$                                   4. $60 m{s}^{-1}$

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

1. ${\left(velocity\right)}^{3/2}$                                         
2. ${\left(dis\tan ce\right)}^2$
3. ${\left(dis\tan ce\right)}^{-2}$                                         
4. ${\left(velocity\right)}^{2/3}$

A bus is moving with a speed of $10$ ${\mathrm{ms}}^{}$ $-1$ on a straight road. A scooterist wishes to overtake the bus in $100$ $\mathrm{s}$. If the bus is at a distance of $\mathrm{1 }\mathrm{km}$ from the scooterist, with what minimum speed should the scooterist chase the bus?

1. 20 ms^-1

2. 40 ms^-1

3. 25 ms^-1                                         

4. 10 ms^-1

The distance travelled by a particle starting from rest and moving with an acceleration $\frac{4}{3}m{s}^{-2},$ in the third second is 

(1) 6m                             

(2) 4m

(3) $\frac{10}{3}m$                         

(4) $\frac{19}{3}m$