In the impoverished mountain kingdom of Pontevedro, the Royal Artillery has deve

Question

In the impoverished mountain kingdom of Pontevedro, the Royal Artillery has developed a method to conserve gunpowder: its cannon is fired horizontally from a cliff, with the muzzle velocity controlled so that the cannonball will strike the target on the plain below on the second bounce. The two Royal Engineers agree that on the first bounce, the ball loses 40% of its vertical speed, but none of its horizontal speed. (a) If the cliff is 1 km high, what muzzle speed is needed to hit a target 9 km from the base of the cliff? (b) If the initial energy of the cannonball is proportional to the mass of gunpowder used, what is the percentage of gunpowder saved by the Royal Bounce method?

Explanation / Answer

h = 1/2 g t1^2

t1 = (2 * 1000 / 9.8)^1/2 = 14.3 sec   time to fall 1000 m

V1 = 9000 / t1 = 630 m/s     horizontal speed needed for 1st case

T = t1 + t2 + t3       time required for second case

T = t1 + 2 t2      since time to reach apex of bounce is same as time to fall

Vy1 = g t = 9.8 * 14.3 = 140 m/s    vertical speed after falling 1000 m

Vy2 = .6 Vy1 = 84 m/s

t2 = Vy2 / g = 84 / 9.8 = 8.57 sec       time to reach apex of bounce

T = 14.3 + 2 * 8.57 = 31.4 sec   time to reach target

Vx2 = 9000 / T = 286 m/s      muzzle speed for second case

E2 / E1 = (Vx2 / Vx1)^2       since KE proportional to V^2

E2 / E1 = (286 / 630)^2 = .206

So 79.4 %   is the savings in gunpowder

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