Introduction to Projectile Motion
Projectile motion is the study of an object moving in two dimensions under the sole influence of gravity (neglecting air resistance). An object launched with an initial velocity $v_0$ at an angle $\theta$ experiences a constant downward acceleration due to gravity, $g$.
The Vectors: Velocity and Acceleration
To determine if the velocity vector $\vec{v}$ and acceleration vector $\vec{a}$ can ever be perpendicular, we must analyze them at any arbitrary point in the flight.
- Acceleration ($\vec{a}$): In projectile motion, the acceleration is always constant and points vertically downward. Therefore, $\vec{a} = (0, -g)$.
- Velocity ($\vec{v}$): The velocity has two components. The horizontal component $v_x$ remains constant ($v_x = v_0 \cos \theta$), while the vertical component $v_y$ changes due to gravity ($v_y = v_0 \sin \theta - gt$). Thus, $\vec{v} = (v_x, v_y)$.
Is Acceleration Ever Perpendicular to Velocity?
Two vectors are perpendicular if their dot product is zero. Mathematically, $\vec{a} \cdot \vec{v} = 0$.
$$\vec{a} \cdot \vec{v} = (0 \times v_x) + (-g \times v_y) = -g v_y$$
For the vectors to be perpendicular, $-g v_y$ must equal zero. Since $g$ (acceleration due to gravity) is a constant ($9.8 \text{ m/s}^2$), this implies that $v_y$ must be zero.
The Result
$v_y = 0$ occurs exactly at the highest point of the trajectory (the peak or apex).
At the peak, the projectile is momentarily moving purely horizontally. Since the acceleration is purely vertical (downward), the angle between the velocity vector (horizontal) and the acceleration vector (vertical) is exactly $90^\circ$.