Instantaneous acceleration

Q. The velocity of the particle is given by $v=3t^2+2t$ in $\text{m/s}$. The acceleration and displacement of the particle as a function of time respectively are
[Take $x=0$ at $t=0$]

1. $(t+2)m/s^2,(3t^3+t^2)m$
2. $(6t+2)m/s^2,(t^3+t^2)m$
3. $(6t^2+2)m/s^2,(t^3+t^2)m$
4. $(t+2)m/s^2,(t^3+t)m$

Q. A particle is moving along a straight-line path according to the relation
$S^2=a{t}^2+2bt+c$
$S$ represents the displacement covered in t seconds and $a,b,c$ are constants. The acceleration of the particle varies as

1. $S^{-3}$
2. $S^{3/2}$
3. $S^{-2/3}$
4. $S^2$

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. 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. 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 motion of a body is given by the equation $\frac{dv\left(t\right)}{dt}=6.0-3v\left(t\right).$ Where v(t) is speed in $\frac{m}{s}$ and t in sec. If body was at rest at t = 0

1. The terminal speed is $3.0\frac{m}{s}$
2. The speed varies with the time as $v\left(t\right)=2(1-e^{-3t})\frac{m}{s}$
3. The speed is $0.1\frac{m}{s}$ when the acceleration is half the initial value
4. The magnitude of the initial acceleration is $0.5\frac{m}{s^2}$

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. 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 $V_0=4m/s$ and $S_o=20m.$ At $S=30m$, what is the acceleration of particle?

1. $0.2m/s^2$
2. $0.4m/s^2$
3. $-0.2m/s^2$
4. $-0.4m/s^2$

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. $0{\text{m/s}}^2$

Q. A particle is moving along a straight-line path according to the relation
$S^2=a{t}^2+2bt+c$
$S$ represents the displacement covered in t seconds and $a,b,c$ are constants. The acceleration of the particle varies as

1. $S^{-3}$
2. $S^{3/2}$
3. $S^{-2/3}$
4. $S^2$

Q. The displacement travelled by a particle in a straight-line motion is directly proportional to $t^{1/2}$, where $t=$ time elapsed. What is the nature of motion?

1. Increasing acceleration
2. Decreasing acceleration
3. Increasing retardation
4. Decreasing retardation

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

1. $3t^2+\cos t$
2. $\frac{t^2}{2}+\sin t$
3. $3t^2+\sin t$
4. $\frac{t^2}{2}-\cos t$

Q. A point moves in a straight line so that its displacement $x$ is given by $x^2=1+t^2$ where, $x$ is in $\text{m}$ and time $t$ is in $\text{s}$. Its acceleration in ${\text{m/s}}^2$ at time $t$ is

1. $\frac{t}{x^3}$
2. $\frac{1}{x^3}$
3. $x-\frac{1}{x^3}$
4. $t+\frac{1}{x^3}$

Q. A body initially moving with a velocity of $5{\text{ms}}^{–1}$, attains a velocity of $25{\text{ms}}^{–1}$ in $5s$. The acceleration of the body is:
(Assume constant acceleration)

1. $8{\text{ms}}^{-2}$
2. $7{\text{ms}}^{-2}$
3. $4{\text{ms}}^{-2}$
4. $3{\text{ms}}^{-2}$

Q. The displacement is given by $x=2t^2+t+5$ in $\text{m}$, the acceleration at $t=5s$ will be

1. $8{\text{m/s}}^2$
2. $12{\text{m/s}}^2$
3. $15{\text{m/s}}^2$
4. $4{\text{m/s}}^2$

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.

A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by $(4t^3-2t)$, where t is in sec and velocity in m/s. What is the acceleration of the particle, when it is 2 m from the origin

1. $28m/s^2$

2. $22m/s^2$

3. $12m/s^2$

4. $10m/s^2$

Q. The displacement of a particle varies with time t, $x=a{e}^{-at}+b{e}^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will

1. Go on decreasing with time
2. Be independent of $\alpha$ and $\beta .$
3. Drop to zero when$\alpha =\beta$
4. Go on increasing with time

Q. A particle is moving along a straight-line path according to the relation

$S^2=a{t}^2+2bt+c$

$S$ represents the displacement covered in $t\text{seconds}$ and $a,b,c$ are constants. The acceleration of the particle varies as

1. $S^{-2/3}$
2. $S^{3/2}$
3. $S^2$
4. $S^{-3}$

Q. The velocity of a particle moving in the positive direction of $x-$ axis varies as $v=10\sqrt{x}$ ($v$ is in ${\text{ms}}^{-1},x$ is in $m$). Assuming that at $t=0,$ particle was at $x=0$. Then,

1. The initial velocity of the particle is zero
2. The initial velocity of the particle is $2.5{\text{ms}}^{-1}$
3. The acceleration of the particle is $2.5{\text{ms}}^{-2}$
4. The acceleration of the particle is $50{\text{ms}}^{-2}$

Q. A particle is moving along x-axis whose instantaneous speed is $v=\sqrt{108-9x^2}$. The acceleration of the particle is

1. $-9x{\text{m/s}}^2$
2. $-18x{\text{m/s}}^2$
3. $\frac{-9x}{2}{\text{m/s}}^2$
4. None of these

Q. The displacement $x$ of a particle along a straight line at time $t$ is given by $x=a_0–a_{1}t+a_2{t}^2$. The acceleration of the particle is:

1. $a_0$
2. $a_1$
3. $2a_2$
4. $a_2$

Q. The velocity- time graph of a body is given in figure. The maximum acceleration in ${\text{ms}}^{-2}$ is

1. 4
2. 3
3. 2
4. 1

Q. The relation between time $t$ and distance $x$ 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 speed $v$ of a car moving on a straight road changes according to equation, $v^2=a+bx$, where $a$ and $b$ are positive constants. Then, the magnitude of acceleration in the course of such motion: ($x$ is the distance travelled) :

1. Increases
2. Decreases
3. Stays constant
4. First decreases and then increases

Q. The graph shows the variation of $\frac{1}{v}$ (where $v$ is the velocity of the particle) with respect to time. Then find the value of acceleration at $t=3\text{sec}$

1. $3{\text{m/s}}^2$
2. $5{\text{m/s}}^2$
3. $1{\text{m/s}}^2$
4. $0.5{\text{m/s}}^2$

Q. The displacement of a particle varies with time t, $x=a{e}^{-at}+b{e}^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will

1. Go on decreasing with time
2. Be independent of $\alpha$ and $\beta .$
3. Drop to zero when$\alpha =\beta$
4. Go on increasing with time

Q. A particle is moving along a curve . Then

1. if its speed is constant, it has no acceleration
2. if its speed is increasing, the acceleration of the particle is along its direction of motion
3. The direction of its acceleration cannot be along the tangent.
4. None of these

Q. The position $x$ of a particle varies with time $t$ as $x=\alpha t^2-\beta t^3$. Which of the following is (are) correct?

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. The acceleration of the particle is zero at $t=\frac{\alpha }{3\beta }$.