Motion in a Plane: Dilemma of the Projectile?

📌 CONCEPT: To ensure the projectile lands at the same point from which it was launched, we need to modify the launch velocity so that the range of the projectile is zero, which means the projectile will return to the starting point after a time interval equal to the time of flight for the entire journey.

📐 RULE / FORMULA: The key principle that applies here is the equation for the range of a projectile: R = (u^2 · sin(2θ)) / g, where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

💡 WORKED EXAMPLE: Let's say the initial velocity is 100 m/s and the angle of projection is 60°. To find the required initial velocity, we can rearrange the equation for range to solve for u: u = √(R · g / sin(2θ)). Plugging in the values, we get u = √((0) · 9.8 / sin(120°)) = 0 m/s, which means the projectile will return to the starting point with zero initial velocity.

⚠️ COMMON MISTAKE: Students often forget to consider the time of flight for the entire journey and calculate the range using the equation R = (u^2 · sin(2θ)) / g, which will give them a non-zero range.