Motion Under Variable Acceleration

Q.

The displacement x of a particle moving in one dimension is related to time t by the equation $t=\sqrt{x}+3$ where x is in metres and t in seconds. The displacement of the particle when its velocity is zero is

1. zero

2. 4 m

3. 1 m

4. 0.5 m

Q.

A particle moves in a straight line with a constant acceleration. It changes its velocity from 10 ms^-1 to 20 ms^-1 while passing through a distance of 135 m in ‘t’ seconds. The value of t is:

1. 12

2. 9

3. 10

4. 1.8

Q. A particle moves a distance $x$ in time $t$ according to equation $x=(t+5{)}^{–1}$. The acceleration of particle is proportional to

1. $(\text{velocity}{)}^{3/2}$
2. $(\text{distance}{)}^2$
3. $(\text{velocity}{)}^{2/3}$
4. $(\text{distance}{)}^{-2}$

Q. The position $x$ of a particle with respect to time $t$ along x-axis is given by $x=9t^2-t^3$, where $x$ is in metres and $t$ in seconds. What will be the position of this particle when it achieves maximum speed along the $+x$ direction?

1. 54 m
2. 81 m
3. 24 m
4. 32 m

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

1. $2\beta v^3$
2. $2\alpha \beta v^3$
3. $2{\beta }^2{v}^3$
4. $2\alpha v^3$

Q.

A particle has initial velocity $(2\hat{i}+3\hat{j})$ and acceleration $(0.3\hat{i}+0.2\hat{j})$ the magnitude of the velocity after $10s$ will be:

1. $9\sqrt{2}units$

2. $5\sqrt{2}units$

3. $9units$

4. $5units$

Q.

An object moving with a speed of 6.25 m/s, is decelerated at a rate given by:

$\frac{dv}{dt}=-2.5\sqrt{v}$

where v is instantaneous speed. The time taken by the object, to come to rest, would be:

1. 2 s
2. 4 s
3. 1 s
4. 8 s

Q.

The position $x$ of a particle varies with time as $x=a{t}^2-b{t}^3$. The acceleration of a particle is zero at time $T$equal to

1. $\frac{2a}{3b}$

2. $\frac{a}{b}$

3. $\frac{a}{3b}$

4. $0$

Q. How acceleration is equal to d^2x / dt^2

Q. A car of mass $m$ starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude $P_0$. The instantaneous velocity of this car is proportional to

1. $t^{-1/2}$
2. $t^{1/2}$
3. $t/\sqrt{m}$
4. $t^2{P}_0$

Q. The acceleration of a particle is increasing linearly with time $t$ as $bt$. The particle starts from the origin with an initial velocity $v_0$ 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.

The velocity of a body depends on time according to equation V=20 + 0.1 $t^2$. the body is undergoing

1. Uniform Acceleration

2. Uniform Retardation

3. Non-Uniform acceleration

4. Zero Acceleration

Q.

The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28.
Obtain the distance traversed by the particle between (a) t = 0 s to 10 s, (b) t = 2 s to 6 s.

What is the average speed of the particle over the intervals in (a) and (b)?

Q. A particle of mass ${10}^{-2}kg$ is moving along the positive x-axis under the influence of a force, $F\left(x\right)=\frac{-k}{2x^2}$, where $k={10}^{-2}N{m}^2$. At time $t=0$, it is at $x=1.0m$ and its velocity $v=0$. The speed of the particle when it reaches $x=05m$ is

Q.

The speed verses time graph for a particle is shown in the figure. The distance travelled (in m) by the particle during the time interval $t=0tot=5s$ will be.

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?

Q.

The acceleration of a particle, starting from rest, varies with time according to the relation a = Kt + C, where K and C are constants of motion. The velocity ‘V’ of the particle after a time ‘t’ will be

1. $Kt+Ct$
2. $\frac{1}{2}K{t}^2+Ct$
3. $\frac{1}{2}(K{t}^2+Ct)$
4. $K{t}^2+\frac{1}{2}Ct$

Q. The position of a particle moving along the $x$-axis is expressed as $x=a{t}^3+b{t}^2+ct+d$. The initial acceleration of the particle is

1. $(a+c)$
2. $2b$
3. $6a$
4. $(a+b)$

Q.

The equation of motion of a particle is given by $s=2t^3–9t^2+12t+1$, where s and t are measured in cm and sec. The time when the particle stops momentarily is

1. $1sec$

2. $2sec$

3. $1,2sec$

4. None of the above

Q. The motion of a particle is defined by the position vector $\overrightarrow{r}=A(\cos t+t\sin t)\hat{i}+A(\sin t-t\cos t)\hat{j}$, where $t$ is expressed in seconds.

The position vector and acceleration vector are perpendicular

1. at $t=\sqrt{2}\text{s}$
2. at $t=1.5\text{s}$
3. at $t=1\text{s}$
4. at $t=0$

Q.

A particle starts moving rectilinearly at time t = 0 such that its velocity ‘v’ changes with time ‘t’ according to the equation $v=t^2-t$ where t is in seconds and v in m/s. Find the time interval for which the particle retards.

1. t = 0.5 s to t = 1 s

2. t = 0 to t = 0.5 s

3. t > 1 s

4. t = 0 to t = 1 s

Q.

A particle is moving along the x-axis with its coordinate with the time $t$ given by $x\left(t\right)=-3t^2+8t+10$meter. Another particle is moving along the y-axis with its coordinate as a function of time given by $y\left(t\right)=5-8t^3$meter. . At $t=1$ second , the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$ . Then $v\left(m{s}^{-1}\right)$ is

Q.

The displacement x of a particle varies with time according to the relation $x=\frac{a}{b}(1-e^{-bt})$. Which of the following statements is correct?

1. At $t=\frac{1}{b}$, the displacement of the particle is $\frac{a}{b}$.

2. The velocity and acceleration of the particle at t = 0 are a and b respectively.

3. The particle will come back to its starting point as $t\to \infty$.

4. None of these

Q. A particle moves with an initial velocity $v_0$ and retardation $\alpha v$, where $v$ is it’s velocity at any time $t$

1. The particle will cover a total distance $\frac{v_0}{\alpha }$
2. The particle will come to rest after a time $\frac{1}{\alpha }$
3. The particle will continue to move for a very long time
4. The velocity of the particle will become $\frac{v_0}{2}$ after a time $\frac{1}{\alpha }$

Q. A body starts from rest at time $t=0$, the acceleration time graph is shown in the figure. The maximum velocity attained by the body will be

1. $65\text{m/s}$
2. $55\text{m/s}$
3. $45\text{m/s}$
4. $35\text{m/s}$

Q. A point moves in a straight line under the retardation $a{v}^2$. If the initial velocity is $u$, the distance covered in '$t$' seconds is

1. $\frac{1}{a}\ln (1+aut)$
2. $a\ln \left(aut\right)$
3. $aut$
4. $\frac{1}{a}\ln \left(aut\right)$

Q. A particle is moving along x-axis whose instantaneous speed is $v^2=108-9x^2$. The acceleration of 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. A ball is dropped on the floor from a height of 10 m. It rebounds to a height of 2.5 m. If the ball is in contact with the floor for 0.01 sec, the average acceleration during contact is

1. $\frac{2100m}{{sec}^2}$ downwards
2. $\frac{1400m}{{sec}^2}$
3. $\frac{2100m}{{sec}^2}$ downwards
4. $\frac{700m}{{sec}^2}$

Q. A particle moves along a straight line and its velocity depends on time as $v=4t-t^2\text{m/s}$. Then for first $5\text{s}$

1. Average velocity is $\frac{25}{3}\text{m/s}$
2. Average speed is $10\text{m/s}$
3. Average velocity is $\frac{5}{3}\text{m/s}$
4. Acceleration is $4{\text{m/s}}^2$ at t = 0

Q. The position of a particle moving along the $x$-axis is expressed as $x=a{t}^3+b{t}^2+ct+d$. The initial acceleration of the particle is

1. $6a$
2. $2b$
3. $(a+b)$
4. $(a+c)$