$A$ particle is moving in a uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and its kinetic energy is halved,find the new value of its angular momentum.

$A$ particle is moving with velocity $\vec v = K(y\hat i + x\hat j)$ where $K$ is a constant. The general equation for its path is

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.

$A$ projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20 \text{ m/s}$ when its direction of motion is $45^{\circ}$ with the horizontal. The initial speed of the projectile is . . . . . . in $\text{m/s}$.

$A$ projectile is fired at an angle of $45^o$ with an initial kinetic energy $K$. What is the kinetic energy of the projectile at its maximum height?

$A$ 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 $m/s$)