Projectile Motion Relative Acceleration

To determine the motion of one projectile as seen from another, we analyze their relative acceleration. This depends on the accelerations of the individual projectiles in a common inertial frame.

Calculating Relative Acceleration

Let $\vec{a}_1$ be the acceleration of the first projectile and $\vec{a}_2$ be the acceleration of the second projectile. The problem states that both projectiles experience the same acceleration:

$ \vec{a}_1 = \vec{a}_2 $

The acceleration of projectile 1 as observed from the frame of reference of projectile 2 is the relative acceleration, $\vec{a}_{12}$:

$ \vec{a}_{12} = \vec{a}_1 - \vec{a}_2 $

Substituting the given condition $\vec{a}_1 = \vec{a}_2$ into the equation:

$ \vec{a}_{12} = \vec{a}_1 - \vec{a}_1 = \vec{0} $

Understanding Relative Motion Dynamics

A relative acceleration of zero ($ \vec{a}_{12} = \vec{0} $) means that the relative velocity between the two projectiles is constant. This implies:

• The distance and orientation between the projectiles remain fixed if their initial relative velocity was zero.
• The projectiles move with a constant relative velocity if their initial relative velocity was non-zero.

In essence, from the perspective of one projectile, the other moves with uniform velocity. Uniform velocity motion results in a straight line path (or a stationary point if relative velocity is zero).

Conclusion on Observed Path

Therefore, the motion of one projectile as seen from another, when both have the same acceleration, is always a straight line.