Velocity, Acceleration & Variable Motion

A particle starts moving from time $t = 0$ and its coordinate is given as $x(t) = 4t^3 - 3t$ A. The particle returns to its original position (origin) 0.866 units later

B. The particle is 1 unit away from origin at its turning point

C. Acceleration of the particle is non-negative

D. The particle is 0.5 units away from origin at its turning point

E. Particle never turns back as acceleration is non-negative

Choose the correct answer from the options given below:

At any time ' $t$ ', the co-ordinates of moving particle are $x = at^2$ and $y = bt^2$. The speed of the particle is

A particle moves along the $x$-axis and has its displacement $x$ varying with time t according to the equation:

$x = c_0\left(t^2 - 2\right) + c(t - 2)^2$

where $c_0$ and c are constants of appropriate dimensions.

Then, which of the following statements is correct?

In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t = x^2 + x$. The acceleration of the particle is

Two cars P and Q are moving on a road in the same direction. Acceleration of car P increases linearly with time whereas car Q moves with a constant acceleration. Both cars cross each other at time t = 0, for the first time. The maximum possible number of crossing(s) (including the crossing at t = 0) is ________.

Two cars start from a point at the same time in a straight line and their positions are represented by $x_1(t) = at + bt^2$ and $x_2(t) = Ft - t^2$. At what time do the cars have the same velocity?

A particle is moving along $x$-axis with its position (x) varying with time $(t)$ as $x = \alpha t^4 + \beta t^2 + \gamma t + \delta$. The ratio of its initial velocity to its initial acceleration, respectively, is:

A particle is moving in a straight line. The variation of position '$x$' as a function of time '$t$' is given as $x = \left(t^3 - 6t^2 + 20t + 15\right)\text{ m}$. The velocity of the body when its acceleration becomes zero is:

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. Its initial position was $16\text{ m}$ right of origin. The variation of its position $(x)$ with time $(t)$ is given as $x = -3t^3 + 18t^2 + 16t$, where $x$ is in $\text{m}$ and $t$ is in $\text{s}$.

The velocity of the particle when its acceleration becomes zero is _________ $\text{m/s}$.

The co-ordinates of a moving particle at any time ' $t$ ' are given by $x = \alpha t^3$ and $y = \beta t^3$ where $\alpha$ and $\beta$ are constants. The speed of the particle at time ' $t$ ' is given by

The relation between time '$t$' and distance '$x$' is $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is:

A particle moves in a straight line so that its displacement $x$ at any time $t$ is given by $x^2 = 1 + t^2$. Its acceleration at any time $t$ is $x^{-n}$ where $n =$ _________.

A particle initially at rest starts moving from reference point $x = 0$ along $x$-axis, with velocity $v$ that varies as $v = 4\sqrt{x}\text{ m/s}$. The acceleration of the particle is __________ $\text{ms}^{-2}$.

Which one of the following statements is Wrong?

The position '$x$' of a particle varies with a time as $x = at^2 - bt^3$ where '$a$' and '$b$' are constants. The acceleration of the particle will be zero at

Two cars A and B start from a point at the same time in a straight line and their positions are represented by $R_A(t) =$ at $+bt^2$ and $R_B(t) = xt - t^2$. At what time do the cars have same velocity?

The position of a particle is given by

$\vec{r}(t) = 4t\hat{i} + 2t^2\hat{j} + 5\hat{k}$

where $t$ is in seconds and $r$ in metre. Find the magnitude and direction of velocity $v(t)$, at $t = 1\text{ s}$, with respect to $x$-axis

The position of a particle related to time is given by $x = \left(5t^2 - 4t + 5\right)\text{ m}$. The magnitude of velocity of the particle at $t = 2\text{ s}$ will be:

The distance travelled by an object in time $t$ is given by $s = (2.5)t^2$. The instantaneous speed of the object at $t = 5\text{ s}$ will be:

The distance travelled by a particle is related to time t as $x = 4t^2$. The velocity of the particle at t=5s is :-

Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $X_P(t) = \alpha t + \beta t^2$ and $X_Q(t) = ft - t^2$. At what time, both the buses have same velocity?

A particle is moving in a straight line such that its velocity is increasing at 5 ms${}^{-1}$ per meter. The acceleration of the particle is _____________ ms${}^{-2}$ at a point where its velocity is 20 ms${}^{-1}$.

If $t = \sqrt{x} + 4$, then $\left(\frac{dx}{dt}\right)_{t=4}$ is:

The instantaneous velocity of a particle moving in a straight line is given as $V = \alpha t + \beta t^2$, where $\alpha$ and $\beta$ are constants. The distance travelled by the particle between 1s and 2s is:

The velocity of a particle is v = v0 + gt + Ft2. Its position is x = 0 at t = 0; then its displacement after time (t = 1) is:

The relation between time t and distance x for a moving body is given as t = mx2 + nx, where m and n are constants. The retardation of the motion is: (When v stands for velocity)

If the velocity of a body related to displacement x is given by $v = \sqrt{5000 + 24x}$ m/s, then the acceleration of the body is .................... m/s2.

The distance x covered by a particle in one dimensional motion varies with time t as x2 = at2 + 2bt + c. If the acceleration of the particle depends on x as x–n, where n is an integer, the value of n is __________

The position of a particle as a function of time t, is given by

x(t) = at + bt2 – ct3

where a, b and c are constants. When the particle attains zero acceleration, then its velocity will be:

The position vector of a particle changes with time according to the relation

$\vec{r}(t) = 15t^2\hat{i} + (4 - 20t^2)\hat{j}$

What is the magnitude of the acceleration at t = 1?

A particle is moving with speed v = b$\sqrt{x}$ along positive x-axis. Calculate the speed of the particle at time t = $\tau$(assume that the particle is at origin t = 0)

Two cars P and Q start from a point at the same time in a straight line and their positions are represented by

xP(t) = (at + bt2) and xQ(t) = (ft $-$ t2).

At what time do the cars have the same velocity?

If the velocity of a particle is v = At + Bt2, where A and B are constants, then the distance travelled by it between 1 s and 2 s is

A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v(x) = \beta x^{-2n}$, where $\beta$ and n are constants and x is the position of the particle. The acceleration of the particle as a function of x, is given by

The displacement 'x' (in meter) of a particle of mass 'm' (in kg) moving in one dimension under the action of a force, is related to time 't' (in sec) by t = $\sqrt{x} + 3$. The displacement of the particle when its velocity is zero, will be

The motion of a particle along a straight line is described by equation x = 8 + 12t $-$ t3 where x is in metre and t in second. The retardation of the particle when its velocity becomes zero is

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

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

The velocity of a particle is v = v0 + gt + ft2. If its position is x = 0 at t = 0, then its displacement after unit time (t = 1) is

The position x of a particle with respect to time t along x-axis is given by x = 9t2 $-$ t3 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?

A particle moving along x-axis has acceleration f, at time t, given by f = f0 $\left(1 - \frac{t}{T}\right),$ where f0 and T are constants. The particle at t = 0 has zero velocity. In the time interval between t = 0 and the instant when f = 0, the particle's velocity (vx) is

A particle moves along a straight line OX. At a time t (in seconds) the distance x (in metres) of the particle from O is given by x = 40 + 12t $-$ t3. How long would the particle travel before coming to rest?

A particle located at x = 0 at time t = 0, starts moving along the positive x-direction with a velocity 'v' that varies as $v = \alpha\sqrt{x}$. The displacement of the particle varies with time as

The displacement x of a particle varies with time t as x = ae$-\alpha$t + be$\beta$t, where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will

The relation between time t and distance x is t = ax2 + bx where a and b are constants. The acceleration is

The co-ordinates of a moving particle at any time 't' are given by x = $\alpha$t3 and y = βt3. The speed of the particle at time 't' is given by

Motion of a particle is given by equation s = (3t3 + 7t2 + 14t + 8) m. The value of acceleration of the particle at t = 1 sec is

The position $x$ of a particle varies with time ($t$) as $x = at^2 - bt^3$. The acceleration will be zero at time $t$ is equal to

The acceleration of a particle is increasing linearly with time $t$ as $bt$. The particle starts from origin with an initial velocity $v_0$. The distance travelled by the particle in time $t$ will be

A particle moves along a straight line such that its displacement at any time $t$ is given by $s = (t^3 - 6t^2 + 3t + 4)$ metres. The velocity when the acceleration is zero is