At what point in the path of a projectile is the speed a minimum?

This problem is based on the concept of projectile motion. When a particle is thrown near the earth’s surface, it travels along a curved path under constant acceleration directed towards the center of the earth surface.

If we have a projectile fired at an angle $\mathit{\theta }$and initial velocity${\mathit{v}}_{\mathbf{0}}$, we can resolve the velocity into horizontal and vertical components.

Formulae:

$V_h=V_0\cos \theta$

$v_v=v_0\sin \theta$

$\left(v_0\cos \theta \right)$In the projectile motion, the horizontal component of the velocity $\left(v_0\cos \theta \right)$remains constant throughout because there is no acceleration in the horizontal direction. But, the vertical component $\left(v_0\sin \theta \right)$ of the velocity varies with time because gravitational acceleration is acting vertical downwards. When the vertical component of the velocity is in the upward direction, it would decrease with time and at some point, it would become zero. This is because gravitational acceleration is in the opposite direction of this vertical component of velocity. During this time horizontal component of the velocity would remain constant. Then, the object would start moving downwards and its velocity would start increasing as now the acceleration and the velocity would be in the same direction. Speed is the magnitude of the velocity. So, it is the maximum height where the object would have least speed as there would be only horizontal component of the velocity and vertical component of the velocity would be zero.