Exercises: Work and energy balances

A probe moves away from the Earth in a radial direction.

1. a) Draw in your notebook the gravitational force on the probe and its displacement, and reason what sign the work done by the gravitational field has during the motion away.
2. b) Write down the relationship between that work and the change in potential energy, $W_{campo} = -\Delta E_p$, and explain its meaning.

You can do it on paper and send a photo with your answer.

A satellite describes a stable circular orbit around the Earth. Look at the diagram:

1. a) Given that the gravitational force is perpendicular to the velocity at every point of the orbit, what is the work done by the gravitational field in one complete orbit?
2. b) Do the kinetic energy, potential energy, and mechanical energy of the satellite change along that orbit? Justify your answer.

To place a satellite into a certain orbit, its potential energy needs to be increased by $\Delta E_p = 5\times10^{9}\ \text{J}$ and, in addition, it must be given an orbital kinetic energy of $E_c = 4\times10^{9}\ \text{J}$.

1. a) Calculate the total energy that must be supplied by the launch vehicle.
2. b) Explain why putting a satellite into orbit does not consist merely of "lifting" it up to that height.

Unit challenge. How can we connect the fall of an object to satellite launches? Become a space mission engineer from home:

1. Home measurement and uncertainty: Drop a small ball from 5 different heights between $0.50\text{ m}$ and $2.00\text{ m}$. Record the fall time with your cell phone (filming in slow motion and analysing the video frame by frame) (3 repetitions per height), associating its uncertainty ($\pm \Delta h$, $\pm \Delta t$). Prepare a table with the mean values and plot $v^2$ versus $h$ to determine the local value of $g$ using a mechanical energy balance ($mgh = \frac{1}{2}mv^2$).

2. Orbital energy balance: Look up the Earth's mean radius ($R_T = 6371\text{ km}$) on the U.S. Geological Survey (USGS) or NASA website and calculate the mass of the Earth $M_T$ from your $g$. Determine the work required to transfer a $3.5\text{ kg}$ nanosatellite from the surface to a circular orbit at an altitude of $400\text{ km}$.

3. Reasoned proposal and trajectories: Write a short report including a table, a graph, and a theoretical discussion: classify the type of motion (elliptical orbit, parabola, or hyperbola) according to the sign of the total mechanical energy ($E_m < 0$, $E_m = 0$, $E_m > 0$) and calculate the escape velocity.

Grading criteria:

• Rigor in experimental data processing, estimation of uncertainties, and graphical fit.

• Correctness in mechanical energy balances and calculation of gravitational work.

• Physical coherence when linking the sign of $E_m$ to the trajectory and gravitational escape.