This animation shows a bomber dropping a bomb to hit a target. To strike the target, the bomb must not be too high or too low when level with the target. In this problem we find the conditions on which a strike on the target is possible .

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A bomber carrying a small, very heavy bomb is flying at speed V a level height H above the ground. The pilot wants to strike directly an enemy dam. The dam is on top of a rocky outcrop. The base of the dam wall is at a height B from sea level and the top of the dam wall at a height T.

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Given certain assumptions on the dynamics of the bomb, show that the bomb will strike the target directly if and only if


B < - 1
2
 D2
V2
 +H < T


Now, a commander wishes to attack a real dam. The base of the dam is located at 100m above sea level and the dam wall is 50m high. The bomber travels at a top speed ot 800 km per hour and must travel below 200m to avoid radar detection. It must also release the bomb at least 1km from the target to avoid guns defending the dam. Given these constraints, can the bomber destroy the dam? What is the highest point above sea level that the bomb can actually strike? Use g = 9.8 ms-2

Discussion / investigation : You made assumptions in the derivation of this result. For a real bomb there will be small corrections due to wind resistance and other factors. How would these affect your conclusions for the previous part?

Extension: Why not try the follow up problem Dam Busters 2 ?



NOTES AND BACKGROUND

Bombing dams and other key military targets was a real mathematical challenge during World War II. Bombs were dropped from planes and then simply fell under gravity, unlike the guided missiles of today. If a target were to be hit accuratly, then the bomb would have to be released from the plane at very particular distances from the target whilst travelling at very specific speeds and heights. The range of defenses of key targets were analysed carefully to try to find a way to deliver the bomb whilst minimising risk to the bomber crew. Since objects fall along parabolas under gravity the problems reduce to finding which parabolas join two points in space and then creating the conditions such the the bomb falls along that particular parabola.

The most famous bombing raid historically involved the creation of a bomb which bounced on water, like a stone skimming across a lake. Some mathematical ideas surrounding this are described in the follow up problem Dam Busters 2.