Instantaneous velocity

Q. A car moves along a straight line whose equation of motion is given by $s=24t+3t^4–2t^3$, where $s$ is in metre and $t$ is in second. The velocity of the car at $t=0$ will be

1. $7\text{m/s}$
2. $9\text{m/s}$
3. $12\text{m/s}$
4. $24\text{m/s}$

Q. Starting from rest a particle moves in a straight line with acceleration $a=\{2+|t-2|{\text{m/s}}^2\}$. What will be the velocity of particle in m/s at the end of $4\text{s}$?

Q. A particle starting from rest undergoes acceleration given by $a=|t–2|{\text{m/s}}^2$ where, $t$ is time in second. Velocity of particle after $4\text{sec}$ is

1. $1\text{m/s}$
2. $2\text{m/s}$
3. $8\text{m/s}$
4. $4\text{m/s}$

Q. PSLV launches a rocket in a gravity free space with constant acceleration $5{\text{m/s}}^2$ along +x direction (see figure). The length of the chamber inside the rocket is $6\text{m}$. At time $t=0$, a ball is thrown from the left end of the chamber in +x direction with a speed $0.7\text{m/s}$ relative to the rocket. At the same time, another ball is thrown in opposite direction to the previous ball with a speed $0.3\text{m/s}$ relative to the rocket. The time in seconds when the two balls hit each other is

Q. The position of the particle is given by $x=(e^t-\cos t)$ in $\text{m}$. The velocity $\left(\text{in m/s}\right)$ of the particle as a function of time $t$ is

1. $e^t-\sin t$
2. $e^t-\sec t$
3. $e^t+\tan t$
4. $e^t+\sin t$

Q. The instantaneous velocity of a particle moving in a straight line is given as $v=\alpha t+\beta t^2$, where $\alpha$ and $\beta$ are constants. The distance travelled by the particle between $1\text{s}$ and $2\text{s}$ is

1. $3\alpha +7\beta$
2. $\frac{3}{2}\alpha +\frac{7}{3}\beta$
3. $\frac{\alpha }{2}+\frac{\beta }{3}$
4. $\frac{3}{2}\alpha +\frac{7}{2}\beta$

Q. Starting from rest a particle moves in a straight line with acceleration $a=\{2+|t-2|{\text{m/s}}^2\}$. What will be the velocity of particle in m/s at the end of $4\text{s}$?

Q. A particle starts from rest. Its acceleration at time $t=0$ is $5{\text{ms}}^{-2}$ which varies with time as shown in figure. The maximum speed of the particle will be

1. $7.5{\text{ms}}^{-1}$
2. $15{\text{ms}}^{-1}$
3. $30{\text{ms}}^{-1}$
4. $7.5{\text{ms}}^{-1}$

Q. The rate change of displacement with time is .

1. speed
2. velocity
3. acceleration

Q. How small is an instant?

1. Very small time interval
2. Zero time interval
3. Infinitesimally small time interval
4. Not defined

Q. Statement -1 : Let $v$ be the speed of particle at any time $t$. If $\frac{dv}{dt}$ is a non - zero constant then speed of particle may be minimum at some time.
Statement -2 : The speed $v$ of a moving particle may be minimum at time $t=t_0$ when $v=0$. The speed $v$ of a moving particle may also be minimum at time $t=t_0$ when $\frac{dv}{dt}=0$ and $\frac{d^{2}v}{d{t}^2}=$ positive.

1. Both statement 1 and statement 2 are correct and statement 2 is the correct explanation of statement 1.
2. Both statement 1 and statement 2 are correct but statement 2 is not the correct explanation of statement 1.
3. Statement 1 is correct and statement 2 is incorrect.
4. Statement 1 is incorrect and statement 2 is correct.