Exercises: The principle of conservation: mechanical energy in everyday situations

The pendulum of an old clock swings between its two highest positions, as shown in the figure.

1. a) At which points along its path is its potential energy maximum? And at which points is its kinetic energy maximum?
2. b) After a few days, the pendulum stops if no one winds it up: explain what role friction plays and what the clock mechanism (the weights) is used for.

A dry branch of $0.5\ \text{kg}$ falls from a tree at a height of $20\ \text{m}$ and falls with negligible friction (take $g = 10\ \text{m/s}^2$).

1. a) What is its potential energy when it falls?
2. b) If mechanical energy is conserved, what kinetic energy will it have when it reaches the ground?
3. c) At what speed will it reach the ground?

Roller coasters only have a motor on the first climb, where a chain pulls the carriages to the top; the rest of the ride is done without a motor. Using energy, explain why the carriage can complete the entire circuit after that first climb.

A coin falls down a vertical well and reaches the bottom at $14\ \text{m/s}$, as shown in the figure. Assume there is no friction and take $g = 9.8\ \text{m/s}^2$. State the conservation of mechanical energy and calculate the depth of the well.

You drop a ball from the top of a roof and, at the same time, allow an identical ball to roll down a frictionless inclined plane starting from the same height, as shown in the figure.

1. a) Which one reaches the ground with greater speed? Reason your answer using the conservation of energy.
2. b) Which one arrives first? Explain the difference between final speed and time of descent.

A boules ball rolls across the grass and eventually comes to a stop.

1. a) Is its mechanical energy conserved?
2. b) What has its initial kinetic energy been converted into?
3. c) Has that energy been destroyed? Explain what the principle of conservation of energy states.

A marble rolls up a toy ramp and stops before reaching the end.

1. a) What happens to its kinetic energy as it goes up? And to its potential energy?
2. b) Explain the energy transformation that takes place.

State in which of these situations mechanical energy is conserved and in which it is not, justifying your answer in one sentence for each case:

1. a) A skydiver descends with an open parachute at a constant speed.
2. b) The Space Station orbits around the Earth.
3. c) A van brakes while going down a mountain pass.

A roller coaster carriage passes over the highest peak of the track, at $45\ \text{m}$, with a speed of $10\ \text{m/s}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) Calculate its mechanical energy per kilogram of mass.
2. b) At what speed will it pass through a point located at a height of $5\ \text{m}$?
3. c) Do you need to know the mass of the carriage to answer? Justify your answer.

A chandelier of $5\ \text{kg}$ swings after a gust of wind: at the ends of the oscillation, it is $0.2\ \text{m}$ higher than at the lowest point (take $g = 10\ \text{m/s}^2$).

1. a) What is its potential energy at one end, relative to the lowest point?
2. b) If there is no friction, at what speed will it pass through the lowest point?

A parcel of $2\ \text{kg}$ goes down a metal ramp from a height of $3\ \text{m}$ and reaches the bottom at $6\ \text{m/s}$, as shown in the figure (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy did it have at the top?
2. b) What kinetic energy does it have when it reaches the bottom?
3. c) Has mechanical energy been conserved? Calculate the dissipated energy and explain where it has gone.

You throw a ball of $0.2\ \text{kg}$ vertically upwards with a speed of $15\ \text{m/s}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What kinetic energy does it have when it leaves your hand?
2. b) What maximum height will it reach? State the conservation of energy.

A skateboarder enters the base of a ramp with a speed of $8\ \text{m/s}$ and goes up it without pushing off, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What maximum height does he reach?
2. b) If he entered at twice the speed, $16\ \text{m/s}$, would he go twice as high? Calculate it and comment on the result.

A skydiver of $80\ \text{kg}$ descends with an open parachute at a constant speed of $5\ \text{m/s}$, from a height of $1000\ \text{m}$ to $800\ \text{m}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate his kinetic energy: does it change during the descent?
2. b) Calculate the potential energy he loses.
3. c) Is his mechanical energy conserved? If not, explain what the lost energy is transformed into.

A box of $5\ \text{kg}$ slides across a kitchen floor with an initial speed of $4\ \text{m/s}$ and stops due to friction, which is equal to $8\ \text{N}$, as shown in the figure.

1. a) What kinetic energy did it have when it started sliding?
2. b) Friction does negative work that consumes all of that energy: calculate the distance it travels before coming to a stop.

A ball of $0.4\ \text{kg}$ is dropped from $1.5\ \text{m}$ and, after bouncing on the ground, only rises to $0.9\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$).

1. a) Calculate its potential energy before falling and after bouncing.
2. b) How much energy has it lost in the bounce? What percentage of the initial energy is this?
3. c) What has that energy been transformed into?

A roller coaster carriage starts from rest at a peak $50\ \text{m}$ high, falls into a valley located at $5\ \text{m}$ and then approaches a second peak of $45\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) At what speed does it pass through the valley?
2. b) At what speed does it pass over the second peak?
3. c) If there were a third peak of $55\ \text{m}$, would it manage to clear it? Justify your answer using energy considerations and state the minimum height it should start from to achieve this.

A pendulum of $1\ \text{kg}$ passes through its lowest point with a speed of $3\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$).

1. a) What kinetic energy does it have at that point?
2. b) If there were no friction, what maximum height would it reach on the other side?
3. c) In reality it only rises $0.40\ \text{m}$: calculate the dissipated energy and explain why the pendulum will eventually stop.

A sandbag of $2\ \text{kg}$ falls from a stationary hot-air balloon at a height of $45\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What is its mechanical energy when it starts falling?
2. b) When it passes a height of $25\ \text{m}$, how much potential energy and how much kinetic energy does it have?
3. c) At what speed does it pass through there?
4. d) At what speed does it reach the ground?

A block of $3\ \text{kg}$ slides down a ramp from a height of $2\ \text{m}$ and then drags along the horizontal ground for $4\ \text{m}$, until it comes to a complete stop, as shown in the figure (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy did it have at the top?
2. b) Reason why all that energy ends up being dissipated by friction.
3. c) Assuming that almost all of it is dissipated on the horizontal section, calculate the average friction force slowing down the block.

On a giant fairground slide, there are two lanes that start at the same height of $12\ \text{m}$ and end at the same point at the bottom, as shown in the figure. Lane A is almost straight ($15\ \text{m}$ path) and lane B meanders ($40\ \text{m}$ path). A person of $60\ \text{kg}$ slides on a mat whose average friction force is $10\ \text{N}$ (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy does the person have at the top?
2. b) Calculate the energy dissipated by friction in each lane.
3. c) In which of the two lanes does the person reach the bottom with more kinetic energy? And with greater speed? Justify your answer.

A car of $1000\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a complete stop; its brakes can exert an average force of $8000\ \text{N}$.

1. a) Calculate its kinetic energy and the minimum braking distance (the work done by the braking force consumes all the kinetic energy).
2. b) Repeat the calculation for an initial speed of $40\ \text{m/s}$.
3. c) What is the relationship between the two distances? Explain what consequence this has for road safety.

A cyclist reaches the bottom of a hill with a height gain of $7.2\ \text{m}$ and stops pedalling (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What minimum speed does he need to reach the top without pedalling?
2. b) If he enters at $10\ \text{m/s}$, what height will he reach before stopping?
3. c) The cyclist and bicycle combined have a mass of $70\ \text{kg}$: how much energy is missing to reach the top if he enters at $10\ \text{m/s}$? Who must supply it and how?

A demolition ball of $800\ \text{kg}$ is released from a height of $5\ \text{m}$ above the lowest point of its swing and strikes the wall as it passes through, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) At what speed does it hit?
2. b) What kinetic energy does it carry upon impact?
3. c) If a ball of $400\ \text{kg}$ were used from the same height, would the impact speed change? And the energy of the blow? Reason why demolition balls are so heavy.

Unit challenge. How much mechanical energy is "lost" in an everyday bounce? Become an energy auditor in your own home.

1. Data collection: Weigh a ball ($m$) and drop it from four different heights ($h \pm 0.5\text{ cm}$). Record the process with your phone to accurately measure the maximum height after the bounce ($h' \pm 1.0\text{ cm}$).

2. Processing and calculations: Create a table with uncertainties. Calculate the initial $E_p$ ($m \times g \times h$, using $g = 9.8\text{ m/s}^2$), the theoretical $E_c$ just before impact by conservation, and the dissipated non-conservative work: $W_{nc} = \Delta E_m = m \times g \times (h' - h)$.

3. Graph and power: Plot $h'$ versus $h$. Estimate the average mechanical power transferred during the collision ($P = |W_{nc}|/\Delta t$, considering $\Delta t \approx 0.02\text{ s}$) and compare it with the power of a lightbulb in your home by checking its technical label or the REE website.

4. Final report: Write a reasoned proposal explaining the degradation of mechanical energy during the impact and how surfaces influence it.

Evaluation criteria:

• Systematic recording of experimental measurements with their uncertainties, table, and consistent linear graph.

• Correct application of the formulas for $W$, $P$, $E_c$, and $E_p$, deducing the $E_m$ balance.

• Quality of the report in the physical justification of energy conservation and contextualization of the result.