3. Two projectiles are thrown with the same initial velocity, one at an angle Theta and the other at an angle of 90^circ -Theta (a) Can both projectiles strike the ground at the same distance from the projection point? (b) Can both projectiles be in air for the same time interval?

(a) Yes, both projectiles can strike the ground at the same distance from the projection point.

(b) No, both projectiles cannot be in air for the same time interval.

Explanation:

(a) For two projectiles to strike the ground at the same distance from the projection point, the horizontal components of their velocities must be equal. Since both projectiles are thrown with the same initial velocity, their horizontal components of velocity will be equal if the angles of projection are complementary, i.e., if one is thrown at an angle $$\Theta$$ and the other at $$90^{\circ} - \Theta$$ . This is because the horizontal component of velocity is given by $$v \cos(\Theta)$$ , where $$v$$ is the initial velocity. Therefore, if $$\Theta$$ and $$90^{\circ} - \Theta$$ are complementary, their cosines will be equal, and the horizontal components of velocity will be equal.

(b) The time of flight of a projectile is given by $$T = \frac{2v \sin(\Theta)}{g}$$ , where $$v$$ is the initial velocity, $$\Theta$$ is the angle of projection, and $$g$$ is the acceleration due to gravity. Since the initial velocities of the two projectiles are the same, their times of flight will be equal only if their angles of projection are equal. However, in this case, the angles of projection are complementary, so their times of flight will not be equal. Therefore, both projectiles cannot be in air for the same time interval.