Aliens have invaded. To stop them, you must design a simple projectile weapon th

Question

Aliens have invaded. To stop them, you must design a simple projectile weapon that fires metal spheres upwards towards the invading alien ships. The basic design is shown in the figure. A compressed linear spring is used to launch a 10 kg metal sphere upwards. The spring is initially compressed 10m. To damage the alien ships, you calculate that the sphere must be travelling upwards at 100 m/s at an elevation of 1000m. Find the launch speed and spring rate required. You may neglect the potential energy due to the height of the launching system in your calculations. Use g=10m/s2. Find the kinetic energy of the sphere at its impact point. Also find the average power provided by the launch spring during the launching event.

Explanation / Answer

By the 3rd equation of motion we have v^2=u^2-2gh =>u^2=v^2+2gh=100^2+2x10x1000=30000 =>u=173.21 a)Launch speed=u=173.21 m/s b)When the spring is compressed, PE=0.5kx^2 When the spring releases fully KE=0.5mu^2 When the spring releases fully PE=mgx From energy balance we have 0.5kx^2=0.5mu^2+mgx =>kx10^2=10x173.21^2+2x10x10x10 =>k=3000+2000=5000 so spring rate, k=5000 N/m c)KE of the sphere at the impact=0.5mv^2=0.5x10x100^2=50000 J=50 kJ

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