The coordinates 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:

$A$ particle is projected from the ground with some initial velocity making an angle of $45^{\circ}$ with the horizontal. If it reaches a height of $7.5 \ m$ above the ground, while it travels a horizontal distance of $10 \ m$ from the point of projection, then the initial speed of the particle is (assume, $g=10 \ m/s^2$): (in $m/s$)

$A$ particle of mass $m$ is executing uniform circular motion on a path of radius $r$. If $p$ is the magnitude of its linear momentum,then the radial force acting on the particle is:

$A$ particle moving in the $xy$ plane experiences a velocity-dependent force $\overrightarrow{F} = k(v_y \hat{i} + v_x \hat{j})$,where $v_x$ and $v_y$ are the $x$ and $y$ components of its velocity $\overrightarrow{v}$. If $\overrightarrow{a}$ is the acceleration of the particle,then which of the following statements is true for the particle?

Range of a bullet fired at $45^o$ to the horizontal is $980 \, m$. If the bullet is fired at the same angle from a car travelling horizontally at $18 \, km/h$ towards the target,then the range will be increased by:

$A$ particle of mass $M$ is moving in a horizontal circle of radius $R$ with uniform speed $v$. When the particle moves from one point to a diametrically opposite point,its