Instantaneous acceleration

Q. A particle moves along a straight line. Its position at any instant is given by $x=32t-\frac{8t^3}{3}$ where $x$ is in $\text{m}$ and $t$ in $\text{s}$. The acceleration of the particle at the instant when particle is at rest will be

1. $-16{\text{m/s}}^2$
2. $-32{\text{m/s}}^2$
3. $16{\text{m/s}}^2$
4. $32{\text{m/s}}^2$

Q. A boy stretches a stone against the rubber tape of a catapult or 'gulel' (a device used to detach mangoes from the tree by boys in Indian villages) through a distance of $24\text{cm}$ before leaving it. The tape returns to its initial normal position, accelerating the stone over the stretched length. The stone leaves the gulel with a speed of $2.2\text{m/s}$. Assuming that acceleration is constant while the stone was being pushed by the tape, find the magnitude.

1. $8.1{\text{m/s}}^2$
2. $9.1{\text{m/s}}^2$
3. $10.1{\text{m/s}}^2$
4. $11.1{\text{m/s}}^2$

Q. Given below are the equations of motion of four particles $A,B,C$ and $D$.
$x_A=6t-3$;
$x_B=4t^2-2t+3$;
$x_C=3t^3-2t^2+t-7$;
$x_D=7\cos 60°-3\sin 30°$
Which of these four particles move with uniform non-zero acceleration?

1. $A$
2. $B$
3. $C$
4. $D$

Q.

The relation between time and distance is $t=\alpha x^2+\beta x$ where $\alpha$ and $\beta$ are constants. The retardation is

1. $2\alpha v^3$

2. $2\beta v^3$

3. $2\alpha \beta v^3$

4. $2{\beta }^2{v}^3$

Q. The acceleration of a particle is increasing linearly with time t as bt . The particle starts from the origin with an initial velocity $v_o$. The distance travelled by the particle in time t will be

1. $v_{0}t+\frac{1}{3}b{t}^2$
2. $v_{0}t+\frac{1}{3}b{t}^3$
3. $v_{0}t+\frac{1}{6}b{t}^3$
4. $v_{0}t+\frac{1}{2}b{t}^2$

Q. Velocity $\left(V\right)$ versus displacement $\left(S\right)$ graph of a particle moving in a straight line is as shown in the figure. Given $V_o=4m/s$ and $S_o=20m.$
Corresponding acceleration $\left(a\right)$ versus displacement $\left(S\right)$ graph of the particle would be

1. 
2. 
3. 
4. 

Q. A car accelerates uniformly from $13{\text{ms}}^{–1}$ to $31{\text{ms}}^{–1}$ while entering the motorway, covering a distance of $220\text{m}$. Then the acceleration of the car will be:

1. $2.9{\text{ms}}^{-2}$
2. $1.8{\text{ms}}^{-2}$
3. $4{\text{ms}}^{-2}$
4. $2.2{\text{ms}}^{-2}$

Q. The position of the particle is given by $x=2t^3-4t^2+5$ in $\text{m}$. The acceleration of the particle at $5\text{sec}$ is

1. $45m/s^2$
2. $48m/s^2$
3. $40m/s^2$
4. $52m/s^2$

Q. Each of the four particles move along $x$ axis. Their coordinate (in meters) as function of time (in seconds) are given by
Particle 1: $x\left(t\right)=3.5-2.7t^3$
Particle 2: $x\left(t\right)=3.5+2.7t^3$
Particle 3: $x\left(t\right)=3.5+2.7t^2$
Particle 4: $x\left(t\right)=3.5-3.4t-2.7t^2$
Which of these particles are speeding up for $t>0$?

1. All four
2. Only 1
3. Only 2 and 3
4. Only 2, 3 and 4

Q. A particle is moving along positive $x$-axis and at $t=0$, the particle is at $x=0$. The acceleration of the particle is a function of time. The acceleration at any time $t$ is given by $a=2(1–[t\left]\right)$ where $\left[t\right]$ is the greatest integer function . Assuming that the particle is at rest initially, the average speed of the particle for the interval $t=0s\text{to}t=4s$ is

1. $1m/s$
2. $0.5m/s$
3. $2m/s$
4. $1.5m/s$

Q. A particle moves along x-axis and its acceleration at any time $t$ is $a=2sin\left(\pi t\right)$, where $t$ is in seconds and $a$ is in $m/s^2$. The initial velocity of particle (at time $t=0$) is $u=0$. The distance travelled (in meters) by the particle from time $t=0$ to $t=1\text{s}$ will be

1. $\frac{2}{\pi }$
2. $\frac{1}{\pi }$
3. $\frac{4}{\pi }$
4. None of these

Q. The velocity $\left(V\right)$ versus displacement $\left(S\right)$ graph of a particle moving in a straight line is as shown in the figure. Given $V_o=4m/s$ and $S_o=20m.$ At what displacement do velocity and acceleration have the same magnitudes?

1. $\text{At}S=5m$
2. $\text{At}S=10m$
3. $\text{At}S=30m$
4. None of these

Q. Velocity $\left(V\right)$ versus displacement $\left(S\right)$ graph of a particle moving in a straight line is as shown in the figure. Given $V_o=4m/s$ and $S_o=20m.$
Corresponding acceleration $\left(a\right)$ versus displacement $\left(S\right)$ graph of the particle would be

1. 
2. 
3. 
4. 

Q. A particle moves along $X-$ axis. It's equation of motion is $x=2t(t-2)$ where $t$ is in $\text{seconds}$. Choose the correct statement(s).

1. The particle moves with a non-uniform acceleration.
2. The particle momentarily comes to rest at $t=1\text{sec}$.
3. The particle performs $\text{SHM}$
4. The given equation represents uniformly accelerated motion for $t>1\text{sec}$

Q. Velocity of a car as a function of time is given by $v\left(t\right)=(4t^3-e^t+\sin t)$ in $\text{m/s}$. The acceleration in $\left({\text{m/s}}^2\right)$ of the car as a function of time $t$ is

1. $12t^2-e^t-\cos t$
2. $t^2-e^t+\cos t$
3. $12t^2-e^t+\cos t$
4. $t^2+e^t+\cos t$

Q. The displacement $x$ of a particle depends on time $t$ as $x=\alpha t^2-\beta t^3.$ Then,

1. The particle will return to its starting point after time $\frac{\alpha }{\beta }$
2. The particle will come to rest after time $\frac{2\alpha }{3\beta }$
3. The initial velocity of the particle was zero but its initial acceleration was not zero
4. No net force will act on the particle at $t=\frac{\alpha }{3\beta }$

Q. Motion described by two different objects are shown in the figure. The ratio of accleration of the objects i.e. $\frac{a_A}{a_B}$ is

1. $3:2$
2. $5:3$
3. $2:3$
4. $3:5$

Q. A Porsche undergoing uniformly accelerated motion can go from $0\text{km}/\text{h}$ to $100\text{km}/\text{h}$ in $10\text{s}$. What is its instantaneous acceleration?

1. $2.77{\text{m/s}}^2$
2. $3.66{\text{m/s}}^2$
3. $-3.66{\text{m/s}}^2$
4. $0{\text{m/s}}^2$