To find the maximum speed of the arrow fired from the bow, we can use the principle of conservation of mechanical energy. The potential energy stored in the bow is converted into the kinetic energy of the arrow when it is released.
The equation for potential energy stored in a spring is:
Potential energy = 1/2 * k * x^2
Where k is the spring constant and x is the displacement of the spring.
Given that the spring constant k is 250 N/m and the displacement x is 0.340 m, we can calculate the potential energy stored in the bow.
Substituting the values into the equation:
Potential energy = 1/2 * 250 N/m * (0.340 m)^2
Next, we equate the potential energy to the kinetic energy of the arrow:
Potential energy = Kinetic energy
1/2 * 250 N/m * (0.340 m)^2 = 1/2 * m * v^2
Where m is the mass of the arrow and v is its velocity.
Given that the mass m is 0.150 kg, we can solve for the velocity v.
Substituting the values into the equation and solving for v:
1/2 * 250 N/m * (0.340 m)^2 = 1/2 * 0.150 kg * v^2
Simplifying the equation gives:
v^2 = (250 N/m * (0.340 m)^2) / 0.150 kg
v^2 ≈ 5.12 m^2/s^2
Taking the square root of both sides, we find:
v ≈ √(5.12 m^2/s^2)
v ≈ 2.26 m/s
Therefore, the maximum speed of the arrow fired from the bow is approximately 2.26 m/s.