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 the figure below. If the spring is compressed a distance of 0.100 m and the gun fired vertically as shown, the gun can launch a 21.0-g projectile from rest to a maximum height of 24.0 m above the starting point of the projectile.

(a) Neglecting all resistive forces, describe the mechanical energy transformations that occur from the time the gun is fired until the projectile reaches its maximum height.

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(b) Neglecting all resistive forces, determine the spring constant.
______________________ N/m

(c) Neglecting all resistive forces, find the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0), as shown in Figure (b).
__________________m/s

Explanation / Answer

Here ,

maximum height , h = 24 m

mass of projectile , m = 0.021 Kg

spring compression , x = 0.1 m

a)

Initial energy as the spring potential energy is converted to the kinetic energy of projectile when it start moving.

the kinetic energy will convert to potential energy.

b)

let the spring constant is k

Using conservation of energy

spring potential energy = final potential energy of ball

0.5 * k * x^2 = m * g * h

0.5 * k * 0.1^2 = 0.021 * 9.8 * 24

k = 987.9 N/m

the spring cosntant is 987.9 N/m

c)

let the speed of the projectile is v

0.5 (m * v^2 ) = 0.5 * k * x^2

0.021 * v^2 = 987.8 * 0.1^2

v = 21.7 m/s

the speed of the projectile when it is moving through the equilibrium position is 21.7 m/s

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