A projectile is launched from ground level at an angle of 30 degrees above the horizontal. Neglect air resistance and consider the motion from just after the moment it is launched to just before the moment it lands on the ground.
At what point are the velocity and the acceleration perpendicular?
The correct answer and explanation is :
Correct Answer:
The velocity and acceleration are perpendicular at the highest point of the projectile’s trajectory.
Explanation (300 words):
When a projectile is launched, its motion can be broken into two independent components: horizontal and vertical. The horizontal velocity remains constant throughout the flight because there is no air resistance and hence no horizontal acceleration. The vertical velocity, however, changes constantly due to the acceleration caused by gravity (which is always directed downward, toward the center of the Earth).
At launch, the projectile has an initial velocity at an angle of 30° to the horizontal. This velocity can be resolved into two components:
- ( v_x = v \cos(30^\circ) ): the horizontal component
- ( v_y = v \sin(30^\circ) ): the vertical component
As the projectile rises, gravity decelerates the vertical component. Eventually, at the highest point of its path, the vertical velocity becomes zero—the object is momentarily moving only horizontally at this instant. However, gravity is still acting downward, so the acceleration vector is still pointing vertically downward (i.e., in the negative y-direction), with magnitude ( g \approx 9.8 \, \text{m/s}^2 ).
At this highest point:
- The velocity vector is purely horizontal.
- The acceleration vector is purely vertical.
Since one vector is completely horizontal and the other completely vertical, they are perpendicular to each other. Mathematically, the dot product of two perpendicular vectors is zero, which also confirms their perpendicularity.
Therefore, the velocity and acceleration vectors are perpendicular at the highest point of the projectile’s motion—this is a unique point during the flight where this condition occurs.
This understanding is critical in physics because it highlights how vector components can help us analyze motion in two dimensions independently.