Final answer:
The maximum height an arrow can reach when fired vertically from a bow that acts like a spring with a constant of 400N/m and a draw distance of 70cm is 200 meters. The maximum horizontal distance cannot be determined without the launch angle or additional information regarding the projectile's motion.
Step-by-step explanation:
The student's question involves calculating the maximum height an arrow can reach when fired vertically from a bow modeled as a spring and the maximum horizontal distance it might travel when launched at an angle. The bow has a spring constant of 400N/m and is used to launch a 50g arrow with a draw distance of 70cm. To solve for the maximum height, we can use the conservation of energy principle, where the initial potential energy stored in the bow (as a spring) is converted into the gravitational potential energy at the maximum height.
First, the initial potential energy in the spring (spring PE) is given by the expression PE = 0.5 * k * x^2, where 'k' is the spring constant and 'x' is the draw distance. Plugging in our values we get PE = 0.5 * 400N/m * (0.70m)^2 = 98 Joules. This energy is then converted to gravitational potential energy (gravitational PE) at the maximum height, given by PE = m * g * h, where 'm' is the mass, 'g' is the acceleration due to gravity (9.8m/s^2), and 'h' is the height. Solving for 'h', we find that h = PE / (m * g) = 98J / (0.05kg * 9.8m/s^2) = 200 meters. Therefore, the maximum height the arrow can reach is 200 meters.
For the bonus part concerning the maximum horizontal distance, we would need to consider the angle of launch and use the kinematic equations for projectile motion. However, without the angle or additional information, this value cannot be specifically determined.