Tennis Ball Trajectory?

📌 CONCEPT: Conic sections, including parabolas, describe the path of projectiles under the sole influence of gravity.

📐 RULE / FORMULA: For a projectile under gravity, the height (h) at time (t) is given by h = u sin(θ)t - (1/2)gt^2, where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity. The horizontal distance (R) is given by R = u cos(θ)t.

💡 WORKED EXAMPLE: Given u = 25 m/s, θ = 60°, and g = 9.8 m/s^2, we find the maximum height (h_max) by setting the derivative of h with respect to t to zero and solving for t, which gives t = u sin(θ)/g = 25 sin(60°)/9.8. Then, h_max = 25 sin(60°)(25 sin(60°)/9.8). Using R = u cos(θ)t for the horizontal distance, we get R = 25 cos(60°)(25 sin(60°)/9.8).

⚠️ COMMON MISTAKE: Students often confuse the angle of projection with the angle of elevation, or neglect to consider the effect of gravity on the projectile's path.