In an arcade game a 0.099 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 244 N/m and is compressed from its equilibrium position by 4 cm, find the speed with which the disk slides across the surface.
Answer in units of m/s
1.986
To find the speed with which the disk slides across the surface, we can use the principle of conservation of mechanical energy. The initial potential energy stored in the spring due to compression will convert completely into kinetic energy of the disk.
First, let's convert the compression distance from centimeters to meters. Since 1 cm = 0.01 m, the compression distance is 4 cm * 0.01 m/cm = 0.04 m.
Next, let's calculate the initial potential energy stored in the spring. The potential energy stored in a spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
Where:
k = spring constant = 244 N/m
x = compression distance = 0.04 m
Potential Energy = (1/2) * 244 N/m * (0.04 m)^2
Potential Energy = 0.1968 J
Now, since the potential energy converts entirely into kinetic energy, we can equate these two quantities and solve for the speed (v):
Potential Energy = Kinetic Energy
0.1968 J = (1/2) * m * v^2
Rearranging the equation and solving for v:
v^2 = 2 * (Potential Energy / m)
v^2 = 2 * (0.1968 J / 0.099 kg)
v^2 = 3.9697 m^2/s^2
Taking the square root of both sides to find v:
v = sqrt(3.9697 m^2/s^2)
v ≈ 1.992 m/s
Therefore, the speed with which the disk slides across the surface is approximately 1.992 m/s.
kx²/2 = mv²/2
v=sqrt(kx²/m)=sqrt(244•16•10⁻⁴/0.099) =...