Spring-loaded toy gun?

Question

A spring-loaded toy gun is used to shoot a ball of mass m= 1.50kg straight up in the air.The spring has spring constant k=677N/m . If the spring is compressed a distance of 25.0 centimeters from its equilibrium position y=0 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.

1) Find v_m the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position y=0 ).

2) Find the maximum height h_max of the ball.

Please help me thank you

Answer 1

1. Use conservation of energy and MKS units

1/2kx^2=1/2mv^2

v=x*sqrt(k/m) ==> v=(0.25)sqrt(677/1.5)=5.3 m/s
g=9.8 m/s^2

s=-1/2gt^2+5.3t

differentiate to find time where s is maximum

ds/dt=-gt+5.3=0 ==> t=5.3/9.8=0.54 s

s=-1/2(9.8)*(0.54)^2+5.3*(0.54)=1.43 m