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

A particle moves so that its position vector is given by $\vec{r}$ = x cos(ωt) î + y sin(ωt) ĵ, where ω is a constant. Which of the following is true?

If $\theta_1$ and $\theta_2$ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip $\theta$ is given by

The $x$- and $y$-coordinates of the particle at any time are $x = 5t - 2t^2$ and $y = 10t,$ respectively, where $x$ and $y$ are in meters and $t$ in seconds. The acceleration of the particle at $t = 2s$ is

A toy car with charge $q$ moves on a frictionless horizontal plane surface under the influence of a uniform electric field $\mathbf{E}.$ Due to the force $q \mathbf{E},$ its velocity increases from $0$ to $6$ m/s in one second duration. At that instant, the direction of the field is reversed. The car continues to move for two more seconds under the influence of this field. The average velocity and the average speed of the toy car between $0$ to $3$ seconds are respectively

When an object is shot from the bottom of a long smooth inclined plane kept at an angle $60^\circ$ with horizontal, it can travel a distance $x_1$ along the plane. But when the inclination is decreased to $30^\circ$ and the same object is shot with the same velocity, it can travel $x_2$ distance. Then $x_1 : x_2$ will be:

The speed of a swimmer in still water is $20$ m/s. The speed of river water is $10$ m/s and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes w.r.t. north is given by:

A car starts from rest and accelerates at $5$ m/s$^2.$ At $t = 4$ s, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at $t = 6$ s?$\newline ($Take $g = 10$ m/s$^2)$

A particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution.$\newline$If this particle were projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it equals $4R.$ The angle of projection, $\theta,$ is then given by:

The displacement-time graphs of two moving particles make angles of $30^\circ$ and $45^\circ$ with the x-axis. The ratio of their respective velocity is:

A ball is projected with a velocity, $10$ m/s, at an angle of $60^\circ$ with the vertical direction. Its speed at the highest point of its trajectory will be:

A bullet is fired from a gun at the speed of 280 m/s in the direction $30^\circ$ above the horizontal. The maximum height attained by the bullet is $( g = 9.8 m/s^{2}, \sin 30^\circ = 0.5):$

A football player is moving southward and suddenly turns eastward with the same speed to avoid an opponent. The force that acts on the player while turning is:

A particle moving with uniform speed in a circular path maintains:

A bob is whirled in a horizontal plane by means of a string with an initial speed of $v$ rpm. The tension in the string is $T.$ If speed becomes $2v$ while keeping the same radius, the tension in the string becomes: