A spring-loaded toy gun is used to shoot a ball of mass m=1.5 kg straight up in the air. The spring has spring constant k=667 N/m. If the spring is compressed a distance of 25.0 centimeters from its equilibrium position and then released, the ball reaches a maximum height (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume that all movement occurs in a straight line up and down along the y axis.
Find the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position y=0).
To find the muzzle velocity of the ball, we can use the principle of conservation of mechanical energy. In this situation, the total mechanical energy of the system is conserved, which means the sum of the kinetic energy and the potential energy remains constant throughout the motion.
Initially, when the spring is compressed by a distance of 25.0 centimeters, the potential energy stored in the spring can be calculated using Hooke's Law:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the compression distance. Plugging in the given values:
Potential Energy = (1/2) * 667 N/m * (0.25 m)^2
= 20.91875 J
At maximum height, when the ball reaches the top of its trajectory, all the potential energy is converted to kinetic energy.
Therefore, the kinetic energy at the maximum height will be equal to the potential energy at the compressed position:
Kinetic Energy = 20.91875 J
We know that:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the ball and v is the velocity. Rearranging the formula, we can solve for v:
v = sqrt((2 * Kinetic Energy ) / m)
Plugging in the values:
v = sqrt((2 * 20.91875 J) / 1.5 kg)
= sqrt(27.891 J/kg)
= 5.2773 m/s
Therefore, the muzzle velocity of the ball is 5.2773 m/s.