The distance x covered by a particle in one dimensional motion varies with time t as $x^2=a{t}^2+2bt+c$. If the acceleration of the particle depends on x as $x^{-n}$, where n is an integer, the value of n is:

A particle moves such that its position vector $\vec{r}(t)=\cos \omega t\hat{i}+\sin \omega t\hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle:

A particle is moving along the x-axis with its coordinate with time '$t$' given by $x(t)=10+8t-3t^2$. Another particle is moving along the y-axis with its coordinate as a function of time given by $y(t)=5-8t^3$. At $t=1s$, the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$. Then $v$ (in  m/s) is ________.

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 at $t=0$)

The position of a particle as a function of time t, is given by $x(t)=at+b{t}^2-c{t}^3$ where a, b and c are constants. When the particle attains zero acceleration, then its velocity will be?

All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.

A car is standing $200m$ behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2m/s^2$ and the car has acceleration $4m/s^2$. The car will catch up with the bus after a time of : 

Two stones are thrown up simultaneously from the edge of a cliff $240m$ high with initial speed $10m/s$ and $40m/s$ respectively.  Which of the following graph best represents the time variation of relative position of the second stone with respect to the first ?
(Assume stones do not rebound after hitting the ground and neglect air resistance, take $g=10m/s^2$)
(The figures are schematic and not drawn to scale.)

A person climbs up a stopped escalator in $60s$. If standing on the same escalator but escalator running with constant velocity, he takes 40 s. How much time is taken by the person to walk up in the moving escalator?

A man with some passengers in his boat, starts perpendicular to the flow of river $200m$ wide and flowing with $2m/s$. Speed of boat in still water is $4m/s$. When he reaches half the width of river, the passengers asked him that they want to reach the point just opposite to the end from where they have started. If the direction due which he must row to reach the required end is $\alpha =2ta{n}^{-1}\left(\frac{1}{x}\right)$. Find $x$.

A boat moves relative to water with a velocity which is $n=2.0$ times less than the river flow velocity. The angle (in degrees) to the stream direction must the boat move to minimize drifting is (120+x) . The value of x is :

Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho }\frac{dp}{dt}\right)$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to

A rocket is moving in a gravity free space with a constant acceleration of $2m{s}^{-2}$ along $+x$ direction (see figure). The length of a chamber inside the rocket is $4m$. A ball is thrown from the left end of the chamber in $+x$ direction with a speed of $0.3m{s}^{-1}$ relative to the rocket. At the same time, another ball is thrown in -x direction with a speed of $0.2m{s}^{-1}$ from its right end relative to the rocket. The time in seconds when the two balls hit each other is