To find the velocity of the arrow immediately after release, we can use the principle of conservation of mechanical energy. The potential energy stored in the bowstring when it is pulled back is converted into kinetic energy of the arrow after release.
Given: Bowstring displacement (x) = 0.75 m Stiffness of the bow (k) = 200 N/m Mass of the arrow (m) = 50 g = 0.05 kg
The potential energy stored in the bowstring can be calculated using the formula:
potential energy = (1/2) * k * x^2
Substituting the given values, we have:
potential energy = (1/2) * 200 N/m * (0.75 m)^2 potential energy = 56.25 J
Since the potential energy is converted into kinetic energy, we have:
kinetic energy = potential energy
The kinetic energy of an object can be calculated using the formula:
kinetic energy = (1/2) * m * v^2
where v is the velocity of the object.
Substituting the known values, we have:
(1/2) * 0.05 kg * v^2 = 56.25 J
Simplifying the equation:
0.025 kg * v^2 = 56.25 J
v^2 = (56.25 J) / (0.025 kg) v^2 = 2250 m²/s²
Taking the square root of both sides, we get:
v = √(2250 m²/s²) v ≈ 47.43 m/s
Therefore, the velocity of the arrow immediately after release is approximately 47.43 m/s.