Final answer:
To find the speed of the arrow when the extension is 10 cm, we can use Hooke's Law and the principle of conservation of mechanical energy. The speed is calculated to be approximately 0.224 m/s.
Step-by-step explanation:
To solve this problem, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the extension or compression of the spring. The equation for Hooke's Law is F = kx, where F is the force, k is the elastic constant, and x is the extension or compression.
Given that the extension before launch is 20 cm and the elastic constant is 50 N/m, we can calculate the force exerted on the arrow as F = (50 N/m)(0.20 m) = 10 N.
To find the speed when the extension is 10 cm, we can use the principle of conservation of mechanical energy. The mechanical energy of the arrow is initially zero (since the initial speed is zero), and it remains zero throughout the trajectory because there is no external work done on the arrow. Therefore, the mechanical energy is conserved, and we can use the equation for mechanical energy: (1/2)mv² = (1/2)kx², where m is the mass of the arrow, v is the velocity, and x is the extension. Plugging in the values, we can solve for v:
(1/2)(0.020 kg)v² = (1/2)(50 N/m)(0.10 m)²
v² = (0.020 kg)(50 N/m)(0.10 m)² / 0.020 kg
v² = 0.050 m²/s²
v = √(0.050 m²/s²) = 0.224 m/s