In the projectile motion of an object,the object reaches its maximum height where its speed is half of its initial speed. Then the ratio between the range and the maximum height of the projectile is

$A$ particle of mass $10\, g$ moves along a circle of radius $6.4\, cm$ with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to $8 \times 10^{-4}\, J$ by the end of the second revolution after the beginning of the motion? .............. $m/s^2$

$A$ body projected along a frictionless inclined plane of angle of inclination $30^{\circ}$ covers a distance of $40\, m$ along the plane. If the body is projected with the same speed at an angle of $30^{\circ}$ with the ground,it will have a range of: (Take $g = 10\, m/s^2$)

$A$ particle is projected from a point at an angle $\theta$ with the horizontal direction. At any time $t$, if $p$ is the linear momentum, $y$ is the vertical displacement, and $x$ is the horizontal displacement, which of the following graphs does not represent the variation of kinetic energy $KE$ of the particle?

$A$ particle has initial velocity $(2\hat{i} + 3\hat{j})$ and acceleration $(0.3\hat{i} + 0.2\hat{j})$. The magnitude of velocity after $10\, s$ will be:

In the following cases,identify the source providing the centripetal force:
$(i)$ Motion of the Earth around the Sun.
$(ii)$ Motion of an electron around the nucleus.
$(iii)$ Motion of a vehicle on a horizontal curved road.