用抛射体击中目标

Hitting Targets with Projectiles

Your friend offers you 100 dollars for slingshotting a rock from the ground onto a platform 100m tall. The platform is 500m away from you. You decide you want to slingshot the rock at a $45^\circ$ angle, what is the velocity of the rock when it lands on the platform?

Equations you need to use: $$R = V_xt_{air}$$ $$X_y = V_{oy}t_{air} + \frac{1}{2}at^2$$

Since , we know that $R = V\cos \theta t_{air}$

$$t_{air} = \frac{R}{V\cos \theta}$$ $$t_{air} = \frac{500}{V\cos 45}$$

Then, we plug in what we know into $X_y = V_{oy}t_{air} + \frac{1}{2}at^2$ and solve for $V$.

$$100 = V\sin(45)(\frac{500}{V\cos 45}) + \frac{1}{2}(-9.8)(\frac{500}{V\cos 45})^2$$ $$100 = 500 - \frac{1225000}{\frac{1}{2}V^2}$$ $$-400(\frac{1}{2}V^2) = -1225000$$ $$V^2 = 6125$$ $$V = 78.26m/s$$