Question

A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, and a launching ramp. The spring has a spring constant K , the ball has a mass M , and the ramp rises a height Y above the table, the surface of which is a height H above the floor.

Initially, the spring rests at its equilibrium length. The spring then is compressed a distance S , where the ball is held at rest. The ball is then released, launching it up the ramp. When the ball leaves the launching ramp its velocity vector makes an angle THETA with respect to the horizontal.


Throughout this problem, ignore friction and air resistance.


1) Calculate Vr , the speed of the ball when it leaves the launching ramp.

Express the speed of the ball in terms of K,S ,M ,Y , G, and/or .H

Answer 1

To calculate the speed of the ball when it leaves the launching ramp, we can use the principle of conservation of mechanical energy. The speed of the ball is given by Vr = sqrt((K*S^2)/M).

Step-by-step explanation:

To calculate the speed of the ball when it leaves the launching ramp, we can use the principle of conservation of mechanical energy. Initially, the ball has only potential energy stored in the compressed spring. When it leaves the ramp, this potential energy is converted into kinetic energy.

The potential energy stored in the compressed spring is given by PE = (1/2)K*S^2, where K is the spring constant and S is the distance the spring is compressed.

The kinetic energy of the ball, when it leaves the ramp, is given by KE = (1/2)M*Vr^2, where M is the mass of the ball and Vr is its velocity.

Since energy is conserved, we can equate the potential energy to the kinetic energy:

(1/2)K*S^2 = (1/2)M*Vr^2

Simplifying the equation, we get:

Vr = sqrt((K*S^2)/M)

Therefore, the speed of the ball when it leaves the launching ramp is given by Vr = sqrt((K*S^2)/M).

Answer 2

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)