# A projectile is launched with an initial speed of Vo at an angle θ above the horizontal. It lands at the same level from which it was launched. What was the average velocity between launch and landing? Explain please because I can't understand this.

Apr 25, 2016

$\text{Average velocity} = {V}_{\circ} \cos \theta$

#### Explanation:

Typical flight of a projectile is as shown in the picture above.
In the problem it is given that initial velocity ${V}_{\circ}$ at an angle $\theta$ above the horizontal. As such inn the picture $\text{U} = {V}_{\circ}$.

This velocity can be resolved into its $x \mathmr{and} y$ components.
Component along $x$ axis, and
Component along $y$ axis$= {V}_{\circ} \sin \theta$

We also know that both $x \mathmr{and} y$ components are orthogonal or perpendicular to each other, therefore can be treated separately.

Maximum height is achieved due to $\sin \theta$ component of the velocity and Horizontal range is achieved due to $\cos \theta$ component.

$\sin \theta$ component.
This component of the velocity decreases due to action of gravity. Becomes zero at the maximum height point. Then increases due to gravity and becomes equal to initial $\sin \theta$ component but in the opposite direction. We have ignored the friction due to air (Drag) acting on the projectile.
Let $t$ be time of flight.
$\text{Average velocity"="Displacement"/"Time of flight}$

It is given that "It lands at the same level from which it was launched", means that displacement in the $y$ axis is $= 0$. From above equation we obtain
$\text{Average velocity} = \frac{0}{t} = 0$ .....(1)

$\cos \theta$ component.
If we ignore air resistance, $\cos \theta$ component of velocity $= {V}_{\circ} \cos \theta$ remains constant throughout the time of flight. Therefore,
$\text{Average velocity} = {V}_{\circ} \cos \theta$ .....(2)

Now to find the Resultant Average velocity we need to add both vectors along $x \mathmr{and} y$ direction. In this instant it is simple as one of the vectors is $= 0$.

Hence, $\text{Average velocity} = {V}_{\circ} \cos \theta$