Motion Under Variable Acceleration

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

Q. A particle is moving along $x$ – axis whose position is given by $x=4-9t+\frac{t^3}{3}$ . Mark the correct statement(s) in relation to its motion.

1. Direction of motion is not changing at any of the instants
2. Direction of motion is changing at $t=3\text{s}$
3. For $0<t<3\text{s}$, the particle is slowing down
4. For $0<t<3\text{s}$ the particle is speeding up

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.

A particle retards from a velocity $v_0$ while moving in a straight line. If the magnitude of deceleration is directly proportional to the square root of the speed of the particle, find its average velocity for the total time of its motion.

1. $\frac{2v_0}{3}$

2. $\frac{-2v_0}{3}$

3. $\frac{v_0}{3}$

4. $\frac{-v_0}{3}$

Q. The position $x$ of a particle varies with time $\left(t\right)$ as $x=a{t}^2-b{t}^3$. The acceleration at time $t$ of the particle will be equal to zero, when $t$ is equal to

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

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. Velocity versus displacement graph of a particle moving in a straight line is as shown in figure. The acceleration of the particle is:

1. Increases linearly with $x$
2. Constant
3. Increases parabolically with $x$
4. None

Q.

A particle moves according to the equation $\frac{dv}{dt}$ = $\alpha$ - $\beta$ v , where $\alpha$ and $\beta$ are constants. Find the velocity as a funtion of time. Assume body starts from rest.

1. v = ($\frac{\beta }{\alpha }$) (1 - $e^{-\beta t}$)

2. v = ($\frac{\beta }{\alpha }$) ($e^{-\beta t}$)

3. v = ($\frac{\alpha }{\beta }$) (1 - $e^{-\beta t}$)

4. v = ($\frac{\alpha }{\beta }$) ($e^{-\beta t}$)

Q.

A parachutist after bailing out falls $50\text{m}$ without friction. When parachute opens, it decelerates at $2{\text{m/s}}^2$. He reaches the ground with a speed of $3\text{m/s}$. At what height, did he bail out?

1. 293 m

2. 111 m

3. 91 m

4. 182 m

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 position of a particle moving in the xy-plane at any time is given by $x=(3t^2-6t)$ metres, $y=(t^2-2t)$ metres. Select the correct statement about the moving particle from the following

1. The acceleration of the particle is zero at t = 0 second
2. The velocity of the particle is zero at t = 0 second
3. The velocity of the particle is zero at t = 1 second
4. The velocity and acceleration of the particle are never zero

Q.

Referring to a - x graph, find the velocity when the displacement of the particle is 100 m. assume initial velocity as zero.

1. 9.8 m/s

2. -10 $\sqrt{6}$ m/s

3. 20 m/s

4. 10 $\sqrt{6}$ m/s

Q. Velocity-time equation of a particle moving in a straight line is, $v=(10+2t+3t^2)$. Find displacement of particle from the origin at time $t=1\text{s}$, if it is given that displacement is $20\text{m}$ at time $t=0$

1. $16\text{m}$
2. $64\text{m}$
3. $48\text{m}$
4. $32\text{m}$

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 to t = 1 s

2. t = 0.5 s to t = 1 s

3. t = 0 to t = 0.5 s

4. t > 1 s

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

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 parallel

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

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 parallel

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

Q. The acceleration of a particle starting from rest varies with time according to the relation $a=kt+c$. The velocity $v$ of the particle after time $t$ will be

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

Q. A particle is moving along $x$ - axis and its coordinate with time is given as $x=-40+5t^2$. Then:

1. Particle will cross the origin at $t=2\sqrt{2}\text{s}$.
2. Magnitude of velocity and acceleration are equal at $t=1\text{s}$
3. Particle will cross the origin at $t=5\sqrt{2}\text{s}$.
4. Magnitude of velocity and acceleration are equal at $t=3\text{s}$.

Q. Given curve is $\frac{1}{v}$ versus x for a particle under motion is, where v is velocity and x is position. The time taken in seconds by particle to move from x = 4 to x = 12 m is (Given - particle is at origin at t = 0)

Q. The position $x$ of a particle varies with time $\left(t\right)$ as $x=a{t}^2-b{t}^3$. The acceleration at time $t$ of the particle will be equal to zero, when $t$ is equal to

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

Q. The retardation experienced by a moving motor boat after its engine is cut off, is given by $\frac{dv}{dt}=-8v^2$. If the magnitude of velocity at cut off is $v_0=3\text{m/s}$, the magnitude of the velocity $1\text{sec}$ after the cut off is (in $\text{m/s}$)

1. $\frac{3}{25}$
2. $\frac{24}{25}$
3. $\frac{1}{8}$
4. $\frac{3}{28}$

Q. The acceleration of a particle starting from rest varies with time according to the relation $a=kt+c$. The velocity $v$ of the particle after time $t$ will be

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

Q.

The velocity of a particle moving along a straight line increases according to the linear law v = $v_0$ + kx

where k is a constant . Then

1) the accelearation of the particle is k($v_0$ + kx)

2) the particle takes a time $\frac{1}{k}$ $lo{g}_e$ ($\frac{v_1}{v_0}$) to attain a velocity $v_1$

3) velocity varies linearly with displacement with slope of velocity displacement curve equal to k

4) data is insufficient to arrive at a conclusion.

1. 1, 2, 3 are correct

2. 2, 3 are correct

3. all of them are correct

4. 1 and 2 are correct

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$2m$from the origin

1. $28m/s^2$
2. $22m/s^2$
3. $12m/s^2$
4. $10m/s^2$

Q. A particle moves along x-axis as $x=4(t-2)+a(t-2{)}^2$ Which of the following is true?

1. The initial velocity of particle is 4
2. The acceleration of particle is 2a.
3. The particle is at origin at t = 0
4. None of these

Q.

An ant starts from the origin and walks along a straight line. After every second, its position (in mm) is observed by a curious 5-year old. These are his readings:

Position (mm) 0 10 40 100 200

After time (s) 0 1 2 3 4

By looking at this table, which of the following could be a possible description of the ant's motion is

(a) Uniform velocity

(b) uniform acceleration (with non-uniform velocity)

(c) Uniform jerk (with non-uniform accelerator)

(d) cannot be determined (can be any of the above)

1. Uniform velocity

2. uniform acceleration (with non-uniform velocity

3. Uniform jerk (with non-uniform accelerator)

4. cannot be determined (can be any of the above)

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.

A body is moving in a straight line, starting its journey from origin. At any instant, its velocity is given by, $K_1{t}^{(K_1-1)}$ when $K_1$ is a constant and t is the time. Find the acceleration of the particle, when it is at a distance s from the origin:

1. $K_1(K_1-1)s^{(1-1/K_1)}$

2. $K_1(K_1-1)s^{(1-2/K_1)}$

3. $K_1(K_1-1)s^{(2-1/K_1)}$

4. $K_1(K_1-1)s^{(3-1/K_1)}$

Q.

If the speedometer(in km/h) of a bike riding along a straight road varies with time as given $V\left(t\right)=3t^2$

What will be the distance it has covered after time t = 2 hr?

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