A projectile is fired from point (0,h) with speed u at angle θ above the horizontal. A straight line
y = h + d + mx,where d>0,is tangent to the projectile trajectory.The condition for tangency is:

Question

A projectile is fired from point (0,h) with speed u at angle θ above the horizontal. A straight line y = h + d + mx,where d>0,is tangent to the projectile trajectory.The condition for tangency is:

G1 Physics · Accepted Answer

The projectile trajectory is y = h + x tanθ - gx²/(2u²cos²θ). For intersection with the line y = h + d + mx, we get h + x tanθ - gx²/(2u²cos²θ) = h + d + mx. Rearrange: [g/(2u²cos²θ)]x² - (tanθ-m)x + d = 0. For tangency, this quadratic in x must have equal roots, so its discriminant is zero: (tanθ-m)² - 4[g/(2u²cos²θ)]d = 0. Therefore, (tanθ-m)² = 2gd/(u²cos²θ). Multiplying by u²cos²θ, u²cos²θ(tanθ-m)² = 2gd. But cosθ(tanθ-m) = sinθ - m cosθ. Hence, u²(sinθ - m cosθ)² = 2gd.

A projectile is fired from point (0,h) with speed u at angle θ above the horizontal. A straight line y = h + d + mx,where d&gt;0,is tangent to the projectile trajectory.The condition for tangency is:

A projectile is fired from point (0,h) with speed u at angle θ above the horizontal. A straight line y = h + d + mx,where d>0,is tangent to the projectile trajectory.The condition for tangency is: