The spring constant of a toy dart gun is 4442 N/m. To the gun the spring is compressed 1.5 cm. The 8 g dart, fired straight upward, reaches a maximum height of 26 m. The acceleration of gravity is 9.81 Determine the magnitude of the energy dissipated by air friction during the dart's ascent. Answer in units of J.
What is the maximum speed the projectile could have when it returns to its starting point? Assume the difference in the frictional energy loss due to the dimished velocity on
the way down compared to its velocity on the way up is negligible. Answer in units of m/s
To determine the energy dissipated by air friction during the dart's ascent, we can use the conservation of mechanical energy. The initial mechanical energy of the system (toy dart gun + dart) is equal to the potential energy at the maximum height.
1. We need to calculate the initial potential energy of the dart when it reaches the maximum height. The formula for potential energy is:
Potential Energy (PE) = mass * gravity * height
Since the mass of the dart is given as 8 g, we need to convert it to kg: 8 g = 0.008 kg.
PE = 0.008 kg * 9.81 m/s^2 * 26 m
PE = 2.03 J
2. To calculate the energy dissipated by air friction, we subtract the potential energy from the mechanical energy of the system:
Energy dissipated = Initial mechanical energy - Potential energy
Since the mechanical energy is equal to the potential energy at the maximum height:
Energy dissipated = 2.03 J - 2.03 J
Energy dissipated = 0 J
Therefore, the magnitude of the energy dissipated by air friction during the dart's ascent is 0 J.
To find the maximum speed of the projectile when it returns to its starting point (assuming negligible frictional energy loss on the way down compared to the way up), we need to calculate the potential energy at the maximum height and convert it to kinetic energy when the dart reaches its starting point.
1. The potential energy at the maximum height is the same as calculated earlier: 2.03 J.
2. To convert the potential energy into kinetic energy at the starting point, we use the formula for kinetic energy:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
We are looking to find the velocity, so we rearrange the equation:
velocity = √(2 * KE / mass)
Since the kinetic energy is equal to the potential energy at the maximum height:
velocity = √(2 * 2.03 J / 0.008 kg)
velocity ≈ √(507.5 m^2/s^2)
velocity ≈ 22.53 m/s
Therefore, the maximum speed the projectile could have when it returns to its starting point is approximately 22.53 m/s.