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Problem 1.162: A projectile is fired into the air at an angle with the ground. It follows the parabolic path as shown in the diagram, landing on the right. Neglect air resistance. At any instant, the projectile has a velocity $\vec{v}$ and an acceleration $\vec{a}$  . Which one or more of the drawings could not represent the directions for $\vec{v}$and $\vec{a}$  at any point on the trajectory?

Solution:

This is projectile motion. When air resistance is neglected, only one force acts on the body. It is the gravitational pull of earth which is always towards the ground in a vertical direction. So the acceleration of the projectile must be always in the downward vertical direction.

Velocity at any instant will be along the direction of motion of the projectile. Therefore it must be tangential to the path.

Keeping these facts in mind, let us examine each of the four cases given to us.

In diagram $1$.

Velocity is inclined and upward. This must represent the upward rise of the projectile just as it starts its motion. See the acceleration vector is vertically downward. So this is a possible situation.

In diagram $2$.

Velocity is perfectly horizontal. This is is possible at the highest point on the flight path. Acceleration vector is vertically downward. So this is a also possible .

In diagram $3$.

Velocity is inclined and downward. This  represents  the downward motion  of the projectile just as it starts falling. Ahe acceleration vector is vertically downward.. This is shown in the following diagram.

In diagram $4$.

In this the velocity vector is horizontal which is possible at the highest point on the path. But acceleration is shown pointed upward. This is not a possible cases.

Out of the four situations, option $4$ is not practically correct.