The launching mechanism of a toy gun consists of a spring of unknown spring cons

Question

The launching mechanism of a toy gun consists of a spring of unknown spring constant, as shown in Figure P5.39a. If the spring is compressed a distance of 0.120 m and the gun fired vertically as shown, the gun can launch a 20.0-g projectile from rest to a maximum height of 20.0 m above the starting point of the projectile. Neglecting all resistive forces, (a) describe the mechanical energy trans- formations that occur from the time the gun is fired until the projectile reaches its maxi- mum height, (b) determine the spring constant, and (c) find the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0), as shown in Figure P5.39b.

Explanation / Answer

a)

as the gun is fired , the spring potential energy due to compression of spring gets converted into kinetic energy of projectile. due to kinetic energy , the projectile moves upward. as the projectile moves Up, the kinetic energy gets converted into potential energy. as the projectile reaches the highest point, all the kinetic energy gets converted into potential energy.

b)

m = mass of projectile = 20 g = 0.020 kg

x = spring compression = 0.120 m

k = spring constant

h = height gained = 20 m

using conservation of energy

spring potential energy = gravitational potential energy of projectile

(0.5) k x2 = mgh

(0.5) k (0.120)2 = (0.020) (9.8) (20)

k = 544.44 N/m

c)

using conservation of energy

spring potential energy = gravitational potential energy of projectile at x = 0 + kinetic energy gained

(0.5) k x2 = mgx + (0.5) m v2

(0.5) (544.44) (0.120)2 = (0.02) (9.8) (0.120) + (0.5) (0.02) v2

v = 19.74 m/s

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