In an arcade game a 0.117 kg disk is shot
across a frictionless horizontal surface by compressing it against a spring and releasing it.
If the spring has a spring constant of
229 N/m and is compressed from its equilibrium position by 5 cm, find the speed with
which the disk slides across the surface.
Answer in units of m/s.
Energy stored in spring = (1/2) k x^2 = .5 * 229 * (0.05)^2 Joules
so
Ke of puck = (1/2) m v^2 = .5 * 0.117 * v^2 Joules
set them equal and solve for v
To find the speed with which the disk slides across the surface, we can use the principle of conservation of mechanical energy.
First, let's determine the potential energy stored in the compressed spring. Since the spring constant is given as 229 N/m and the spring is compressed by 5 cm (which is equal to 0.05 m), the potential energy stored in the spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Substituting the given values into the formula:
Potential Energy = (1/2) * 229 N/m * (0.05 m)^2
= 0.2875 J
Next, the potential energy is converted into kinetic energy as the disk is released and slides across the surface. The kinetic energy can be calculated using the formula:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the disk and v is its velocity.
Since the potential energy is equal to the kinetic energy in this case (due to the absence of friction and other energy losses), we can equate the two equations:
0.2875 J = (1/2) * 0.117 kg * v^2
Solving for v:
v^2 = (2 * 0.2875 J) / 0.117 kg
v^2 = 4.914 J/kg
v ≈ √4.914 ≈ 2.216 m/s
Therefore, the speed with which the disk slides across the surface is approximately 2.216 m/s.