Acceleration

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 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.

Can the velocity be 0 but acceleration be non zero?

1. Yes

2. No

3. In space
4. More information needed

Q. A train accelerates from $36\text{km/h}$ to $54\text{km/h}$ in $5\text{s}$ then average acceleration of the train is

1. $1{\text{m/s}}^2$
2. $2{\text{m/s}}^2$
3. $0.5{\text{m/s}}^2$
4. $1.5{\text{m/s}}^2$

Q.

This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : An object may have acceleration, even when it has uniform speed.
A : एक वस्तु का तब भी त्वरण हो सकता है, जब इसकी चाल एकसमान हो।

R : If velocity is constant, then there is no change in direction of motion.
R : यदि वेग नियत है, तब गति की दिशा में कोई परिवर्तन नहीं होता है।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. The motion of a body is given by the equation $\frac{dv}{dt}=6-3v$, where $v$ is speed in $m/s$ and $t$ is time in seconds. If the body is at rest at $t=0$, then

1. the terminal speed is $2m/s$.
2. the speed varies with the time as $v=2(1-e^{-3t})m/s$.
3. the speed is $0.1m/s$ when the acceleration is half the initial value.
4. the magnitude of the initial acceleration is $6m/s^2$.

Q. Velocity of the particle in a rectilinear motion is given as $v=3t^4-5t$ in $\text{m/s}$. Find
(i) Acceleration at $2$ sec.
(ii) Average acceleration in $2$ sec.

1. (i) $71m/s^2$
   (ii) $17m/s^2$
2. (i) $81m/s^2$
   (ii) $18m/s^2$
3. (i) $91m/s^2$
   (ii) $19m/s^2$
4. (i) $61m/s^2$
   (ii) $16m/s^2$

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. Instantaneous velocity of an object (starting from origin) varies with time as $v=\alpha -\beta t^2$(where $\alpha$ and $\beta$ are positive constants). If $x$ and $a$ are the position and acceleration of the object as a function of time. Then,

1. $x=\alpha t-\frac{1}{3}\beta t^3$
2. $x=\alpha t-\frac{1}{2}\beta t^2$
3. $\alpha =-\beta t$
4. Maximum positive displacement from origin $=\frac{2\alpha }{3}\sqrt{\frac{\alpha }{\beta }}$

Q. A particle moving eastwards with $5{\text{ms}}^{-1}$. In $10\text{s}$ the velocity changes to $5{\text{ms}}^{-1}$ northwards. The average acceleration in this time is

1. $\frac{1}{\sqrt{2}}{\text{m/s}}^2$ North-East.
2. $\frac{1}{2}{\text{m/s}}^2$ North-West.
3. $\frac{1}{\sqrt{2}}{\text{m/s}}^2$ North-West.
4. Zero.

Q. A particle starts from $x=0$ and moves along a straight line such that its velocity $v$ depends on position $x$ as $v=x+4$.
Acceleration of the particle at $t=0$ is

1. $e^{4}m/s^2$
2. $4e^{4}m/s^2$
3. $4m/s^2$
4. $0$

Q. Acceleration of the particle in a rectilinear motion is given as $a=4t+3$ in $\left(m/s^2\right)$. The velocity and displacement as a function of time $t$ respectively are
(Given that at $t=0,u=3m/s,x=0m$)

1. $(t^2+3),(\frac{t^3}{3}+\frac{3t^2}{2}+3t)$
2. $(t^2+3t),(\frac{t^3}{3}+\frac{3t^2}{2})$
3. $(2t^2+3t+3),(\frac{2t^3}{3}+\frac{3t^2}{2}+3t)$
4. $(2t^2+3t),(\frac{2t^3}{3}+\frac{3t^2}{2})$

Q. For a particle moving along $x$ - axis, the acceleration $a$ of the particle in terms of its $x$- coordinate $x$ is given by $a=-9x$, where $x$ is in meters and $a$ is in ${\text{m/s}}^2$. Take acceleration, velocity and displacement in positive $x$ - direction as positive. The initial velocity of particle at $x=0$ is $u=+6\text{m/s}$. The velocity of particle at $x=2\text{m}$ will be

1. $+6\sqrt{2}\text{m/s}$
2. $-6\sqrt{2}\text{m/s}$
3. $72\text{m/s}$
4. $0$

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. Match Column - $\text{I}$ with Column -$\text{II}$
$\begin{array}{cc}\text{Column - I}& \text{Column - II}\\ \text{i.Velocity of a particle following}& \text{(p)}\alpha \\ \text{equation}x=u(t-8)+\alpha (t-2{)}^2& \\ \text{at}t=0& \\ \text{ii.Acceleration of a particle moving according to equation}& \text{(q)}2\alpha \\ x=u(t-8)+\alpha (t-2{)}^2\text{at}t=0& \\ \text{iii.Velocity of the particle moving}& \text{(r) u}\\ \text{according to equation}x=ut+& \\ 5\alpha t^2\ln (1+\frac{t}{2})+\alpha t^2\text{at}t=0& \\ \text{iv.Acceleration of the particle moving}& \text{(s)}u-4\alpha \\ \text{according to the equation}x=ut+& \\ 5\alpha t^2(1+\frac{t}{2})+\alpha t^2\text{at}t=0& \end{array}$

1. $\text{i}\to \left(s\right);\text{ii}\to \left(q\right);\text{iii}\to \left(r\right);\text{iv}\to \left(q\right)$
2. $\text{i}\to \left(s\right);\text{ii}\to \left(q\right);\text{iii}\to \left(s\right);\text{iv}\to \left(p\right)$
3. $\text{i}\to \left(r\right);\text{ii}\to \left(p\right);\text{iii}\to \left(s\right);\text{iv}\to \left(q\right)$
4. $\text{i}\to \left(r\right);\text{ii}\to \left(r\right);\text{iii}\to \left(r\right);\text{iv}\to \left(q\right)$

Q. Acceleration of the particle in a rectilinear motion is given as $a=2t-6$ in $\left({\text{m/s}}^2\right)$. The displacement as a function of time $t$ respectively is
(Given, at $t=0,u=2\text{m/s},x=2\text{m}$)

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

Q. A particle moves in XY plane such that its position, velocity and acceleration are given by $\overrightarrow{r}=x\hat{i}+y\hat{j};\overrightarrow{v}=v_x\hat{i}+v_y\hat{j};\overrightarrow{a}=a_x\hat{i}+a_y\hat{j}$
which of the following condition is correct if the particles speed is reducing?

1. $x{v}_x+y{v}_x<0$
2. $x{v}_x+y{v}_y+y{v}_y>0$
3. $a_x{v}_x+a_y{v}_y<0$
4. $a_x{v}_x+a_y{v}_y>0$

Q. The plot of velocity versus time graph for a particle moving along a straight line is shown.

The value of dot product of velocity and acceleration of particle at $t=2s$ is

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

Q. The displacement $x$ of a particle along a straight line at time $t$ is given by $x=a_0+a_{1}t+a_2{t}^3$. The acceleration $a$ of the particle at time $t=5$ is

1. $15a_2$
2. $30a_2$
3. $a_1+30a_2$
4. $6a_2$

Q. The $x$ coordinates of a particle at any time $t$ are given by $x=7t+4t^2$ where $x$ is in $m$ and $t$ in $\text{s}$. The acceleration of the particle at $5\text{s}$ is

1. zero
2. $8{\text{m/s}}^2$
3. $20{\text{m/s}}^2$
4. $40{\text{m/s}}^2$

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. Acceleration of the particle in a rectilinear motion is given as $a=2t-2$ in $\left(m/s^2\right)$. The final position of the particle as a function of time $t$ is
(Given that at $t=0,u=2m/s,x=4m$)

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

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.

A car is moving along a straight line. It's displacement (x) - time (t) graph is shown. Match the entries in column I with points on graph as in Column-II.

Column I

(p) x $\to$ negative, v $\to$ positive, a $\to$ positive (q) x $\to$ positive, v $\to$ negative, a $\to$ negative

(r) x $\to$ negative, v $\to$ negative, a $\to$ positive (s) x $\to$ positive, v $\to$ positive, a $\to$ negative.

Column II

1. (p - d) (q - b) (r - c) (s - a)

2. (p - b) (q - d) (r - c) (s - a)

3. (p - a) (q - c) (r - b) (s - d)

4. (p - c) (q - a) (r - d) (s - b)

Q. If the pulley is massless and moves with an upward acceleration $a_o$, find the acceleration of block $m_1$ w.r.t elevator.

1. $\frac{(m_2-m_1)(g-a_o)}{(m_1+m_2)}$
2. $\frac{(m_2-m_1)(g+a_o)}{(m_1+m_2)}$
3. $\frac{(m_2-m_1)g}{m_1}$
4. $\frac{(m_2-m_1)a_o}{m_1}$

Q. Acceleration of the particle in a rectilinear motion is given as $a=4t+3$ in $\left(m/s^2\right)$. The final position of the particle as a function of time $t$ is
(Given, at $t=0,u=3m/s,x=2m$)

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

Q. A car start from rest and acquire a velocity of $54\text{km/h}$ in $2\text{sec}$. Find the acceleration of the car.

1. $5{\text{m/s}}^2$
2. $7.5{\text{m/s}}^2$
3. $15{\text{m/s}}^2$
4. $7{\text{m/s}}^2$

Q. The position $x$ of a particle varies with time $t$ as $x=a{e}^{-\alpha t}+b{e}^{\beta t}$, 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. The velocity displacement curve for an object moving along a straight line is shown in the given figure. At which of the marked points, the object is speeding up?

1. 1
2. 2
3. 1 and 3
4. 1, 2 and 3

Q. The pulley arrangements shown in figure are identical. The mass of the rope being negligible. In case-$\text{I}$, the mass $m$ is lifted by attaching a mass $2m$ to the other end of the rope. In case-$\text{II}$, the mass $m$ is lifted by pulling the other end of the rope with constant downward force $F=2mg$, where $g$ is acceleration due to gravity. The acceleration of mass $m$ in case-$\text{I}$ is

1. more than that in case II

2. equal to that in case II

3. Zero

4. less than that in case II