Final answer:
To find the speed of the ball, Vr, when it leaves the launching ramp, use the conservation of energy principle to set the initial potential energy in the compressed spring equal to the kinetic and gravitational potential energy when the ball leaves the ramp, and then solve for Vr.
Step-by-step explanation:
To calculate the speed of the ball when it leaves the launching ramp, which we will call Vr, we can apply the conservation of energy principle. Because we are instructed to ignore friction and air resistance, the mechanical energy of the system is conserved. This means that the potential energy stored in the compressed spring is converted into the kinetic energy of the ball and the gravitational potential energy gained by the ball as it moves up the ramp to height Y.
The potential energy stored in the spring is given by the equation (1/2)KS2, where K is the spring constant and S is the compression distance. When the ball leaves the ramp, its kinetic energy is given by (1/2)MVr2, and the gravitational potential energy gained is MgY, where M is the mass of the ball, g is the acceleration due to gravity (9.81 m/s2), and Vr is the speed of the ball as it leaves the ramp.
Setting the initial spring potential energy equal to the sum of the kinetic and gravitational potential energy at the ramp's top gives us:
(1/2)KS2 = (1/2)MVr2 + MgY
Solving for Vr yields:
Vr = sqrt((KS2 - 2MgY) / M)