Answer :
The rocket reached a maximum height of 122 m with air resistance doing -880 J of work. Without air resistance, this energy could have been used to gain additional height. Hence, in a friction-free environment, the rocket's maximum height would have been greater than 122 m.
A 3.65-kg model rocket is launched straight up and reaches a maximum height of 122 m with air resistance performing -880 J of work on the rocket. To find out the maximum height the rocket would have achieved without air resistance, we can apply the work-energy principle between the event of the engine cutting out and the rocket reaching its maximum height. Since air resistance did -880 J of work, it means that the rocket had 880 J less energy to climb further up.
Without air resistance, the rocket would have 880 J more energy to convert into potential energy at the maximum height. The potential energy (gravitational potential energy) at height H with earth's gravity (g = 9.81 m/s2) is given by the equation PE = mgh, where m is mass and h is height. If we add the additional 880 J to the maximum height scenario ignoring air resistance, we have:
mgh + 880 J = mgH
Substituting the known values:
3.65 kg * 9.81 m/s2 * 122 m + 880 J = 3.65 kg * 9.81 m/s2 * H
Solving for H yields a new maximum height which is greater than 122 m. By performing these calculations, we can determine the maximum height H that the rocket would reach in the absence of air resistance, which would be the sum of the height achieved with air resistance plus the additional height that 880 J of energy can provide.
Thus, the rocket would have reached a height greater than 122 m in a friction-free environment.
