A cannon is made with a spring placed in a tube. The spring constant of the spring is 24 N/m. A ball of mass 100 g is placed on it and the spring is compressed. The distance by which the spring should be pressed down with the ball so that...

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A cannon is made with a spring placed in a tube. The spring constant of the spring is 24 N/m. A ball of mass 100 g is placed on it and the spring is compressed. The distance by which the spring should be pressed down with the ball so that the velocity of the ball when it released is 20 m/s has to be determined.

When a spring with a spring constant k is displaced by a distance x, the energy stored in it is (1/2)*k*x^2. When the spring in the cannon described in the problem is released the energy stored is converted to kinetic energy of the ball. The kinetic energy of a mass m moving at a velocity v is (1/2)*m*v^2. If a 100 g ball has a velocity of 20 m/s its kinetic energy is (1/2)*0.1*20^2 = 20 J

Substituting the values given:

(1/2)*24*x^2 = 20

=> x^2 = 40/24

=> x^2 = 1.67

=> x = 1.29 m

The spring would have to be compressed by 1.29 m if the velocity of the ball when it is released has to be 20 m/s.