The position of a particle as a function of time \(t\), is given by;
\(x(t)=a t+b t^2-c t^3\)
where \(a\), \(b\) and \(c\) are constants. When the particle attains zero acceleration, then its velocity will be:
1. \( a+\frac{b^2}{4 c} \)
2. \( a+\frac{b^2}{c} \)
3. \( a+\frac{b^2}{3 c} \)
4. \( a+\frac{b^2}{2 c}\)

A scooter accelerates from rest for time t₁ at constant rate a₁ and then retards at constant rate a₂ for time t₂ and comes to rest. The correct value of $\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}$ will be :

1. $\frac{a_1+a_2}{a_2}$

2. $\frac{a_2}{a_1}$

3. $\frac{a_1}{a_2}$

4. $\frac{a_1+a_2}{a_1}$

The displacement x of a particle along a straight line at time t is given by $\mathrm{x}={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{t}+{\mathrm{a}}_2{\mathrm{t}}^2$. The acceleration of the particle is

(1)  ${\mathrm{a}}_0$

(2)  ${\mathrm{a}}_1$

(3)  $2{\mathrm{a}}_2$

(4)  ${\mathrm{a}}_2$

The acceleration $\mathrm{\alpha }$ of a particle starting from rest varies with time according to relation $\mathrm{a}=\mathrm{\alpha t}+\mathrm{\beta }$. The velocity of the particle after a time t will be

1.  $\frac{{\mathrm{\alpha t}}^2}{2}+\mathrm{\beta }$

2.  $\frac{{\mathrm{\alpha t}}^2}{2}+\mathrm{\beta t}$

3.  ${\mathrm{\alpha t}}^2+\frac{1}{2}\mathrm{\beta t}$

4.  $\frac{\left({\mathrm{\alpha t}}^2+\mathrm{\beta }\right)}{2}$

A particle starts from rest at t=0 and undergoes an acceleration a in ms^-2 with time t in second which is as shown

Which one of the following plot represents velocity v in ms^-1 versus time t in second?

1.  

2.  

3.  

4.  

The position x of a particle varies with time (t) as $x=<mspace\; width="1em"></mspace>a{t}^2-b{t}^3$. The acceleration at time t of the particle will be equal to zero, where t is equal to

1. $\frac{2a}{3b}$

2. $\frac{a}{b}$

3. $\frac{a}{3b}$

4. zero

The motion of a particle along a straight line is described by the equation \(x = 8+12t-t^3\) where \(x \) is in meter and \(t\) in seconds. The retardation of the particle, when its velocity becomes zero, is:

The position x of particle varies with time t as $\mathrm{x}={\mathrm{at}}^2-{\mathrm{bt}}^3$. The acceleration of the particle will be zero at time equal to

(1)  $\frac{\mathrm{a}}{\mathrm{b}}$

(2)  $\frac{2\mathrm{a}}{\mathrm{b}}$

(3)  $\frac{\mathrm{a}}{3\mathrm{b}}$

(4)  Zero