Free Fall

On the Moon, where there is no air, an astronaut dropped a hammer and a feather at the same time and both reached the ground at the same time. The scene is shown in the figure.

1. a) What does this fact show about falling objects when there is no air resistance?
2. b) What is the acceleration called with which objects fall on Earth, and what is its approximate value?

In free fall, the speed of an object increases by $9.8\ \text{m/s}$ every second. An object is dropped from rest.

1. a) Calculate its speed at $1\ \text{s}$, $2\ \text{s}$ and $3\ \text{s}$.
2. b) What acceleration does the object have during the fall, and which letter is used to represent it?

A stone is dropped from rest. With $g = 9.8\ \text{m/s}^2$, calculate its velocity after $2\ \text{s}$ of falling using $v = v_0 + g \times t$.

An object is dropped from rest. With $g = 9.8\ \text{m/s}^2$, calculate the distance it has fallen after $3\ \text{s}$ using $h = \frac{1}{2} g \times t^2$.

A ball is dropped from a bridge $20\ \text{m}$ high, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) How long does it take to reach the water?
2. b) With what speed does it arrive?

An object is dropped from a bridge $80\ \text{m}$ high. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the time of fall.
2. b) Calculate the velocity upon reaching the water, in $\text{m/s}$.
3. c) Express this velocity in $\text{km/h}$ and comment on why jumping from a bridge is dangerous even into water.

A ball is thrown vertically downwards with an initial velocity of $5\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) What is its velocity after $2\ \text{s}$?
2. b) What distance has it covered in those $2\ \text{s}$?

A ball is thrown vertically upwards at $15\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) How long does it take to reach its maximum height? Hint: at the highest point, the velocity is zero.
2. b) What maximum height does it reach above the release point?

A stone is thrown vertically upwards from a bridge at $10\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) What maximum height does it reach above the level of the bridge?
2. b) How long does it take to pass the level of the bridge again?

You throw a ball vertically upwards at $20\ \text{m/s}$ from the ground, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) What maximum height does it reach?
2. b) How much time passes until it returns to your hands?
3. c) With what velocity does it return? Compare it with the launch velocity and comment on the result.

A ball is thrown vertically upwards at $25\ \text{m/s}$ from the ground, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the maximum height and the time at which it reaches it.
2. b) Set up the equation to find at what times it passes through the height of $20\ \text{m}$ and solve it: you will get two solutions.
3. c) Calculate the velocity of the ball for each of these two solutions.
4. d) Explain the physical meaning of the two solutions and the relationship between the two velocities.

A stone is dropped from the top of a tower $45\ \text{m}$ high. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the time of fall.
2. b) Calculate the speed with which it hits the ground.

An object is thrown vertically downwards from a window at $8\ \text{m/s}$ and falls from a height of $36\ \text{m}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) Set up the equation $36 = 8 \times t + 5 \times t^2$ and solve it to find the time of fall.
2. b) With what velocity does it hit the ground?

A humanitarian aid package is dropped from a hovering helicopter at a height of $125\ \text{m}$, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) How long does it take to reach the ground?
2. b) With what velocity does it arrive?
3. c) Calculate how far it falls in the final second by comparing the distances fallen after $4\ \text{s}$ and after $5\ \text{s}$.

To measure reaction time, one person holds a vertical ruler by its top end and drops it without warning; another person catches it after it has fallen $20\ \text{cm}$, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) Express the distance in meters and calculate the time the ruler has been falling.
2. b) If a person's typical reaction time is $0.25\ \text{s}$, what does the result say about the reflexes of the person who caught it?

Some keys drop from a balcony at a height of $5\ \text{m}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) How long do they take to reach the ground and with what velocity do they arrive?
2. b) From what height would they have to fall for the time of fall to double? Is it also double the height? Comment on the result.

From the edge of a roof $40\ \text{m}$ above the ground, a ball is thrown vertically upwards at $10\ \text{m/s}$, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) What maximum height does it reach above the roof? And above the ground?
2. b) How long does it take to go up?
3. c) Set up the position equation taking the ground as the origin and calculate the total time of flight until the ball reaches the ground.
4. d) With what velocity does it hit the ground?

From the top of a tower $80\ \text{m}$ high, a stone is dropped, and at the same instant, another stone is thrown vertically upwards from the ground at $40\ \text{m/s}$, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) Write the height equations for each stone, $h_1$ and $h_2$, measured from the ground.
2. b) Calculate the instant at which they cross paths.
3. c) At what height do they cross paths?
4. d) Calculate the velocity of each stone at that instant and state whether each one is moving up or down.

A hot air balloon ascends at a constant velocity of $5\ \text{m/s}$. When it is $30\ \text{m}$ above the ground, a ballast bag comes loose. Take $g \approx 10\ \text{m/s}^2$.

1. a) Explain why the bag, upon coming loose, continues to move upwards for a moment instead of falling immediately.
2. b) Taking the ground as the origin and upwards as positive, set up the position equation of the bag and calculate the time it takes to reach the ground.
3. c) With what velocity does it hit the ground?

There is an urban legend that a coin dropped from a skyscraper $180\ \text{m}$ high can be fatal. Take $g \approx 10\ \text{m/s}^2$ and first assume there is no air resistance.

1. a) Calculate the time of fall.
2. b) Calculate the velocity with which it would hit the ground and express it in $\text{km/h}$.
3. c) In reality, air resistance slows down such a light object significantly. Reason whether the actual speed will be greater or smaller than the calculated one and which part of the free fall model no longer holds.

A diver drops from a diving board at a height of $19.6\ \text{m}$.

1. a) With $g = 9.8\ \text{m/s}^2$, calculate the time of fall and the speed of entry into the water.
2. b) Repeat the calculation using the approximation $g \approx 10\ \text{m/s}^2$, rounding to two decimal places.
3. c) Compare the results and comment on whether the approximation $g \approx 10\ \text{m/s}^2$ overestimates or underestimates the time of fall, and in what type of problems it might be acceptable.

Two balls are dropped from a height of $45\ \text{m}$, the second one second after the first. Take $g \approx 10\ \text{m/s}^2$.

1. a) How long does the first ball take to reach the ground?
2. b) At the instant the first ball reaches the ground, for how long has the second ball been falling and at what height is it?
3. c) Calculate the distance separating the balls when the first has been falling for $2\ \text{s}$ and when it has been falling for $3\ \text{s}$. Does the distance between them increase or decrease as they fall? Justify your answer.

In a school "egg drop" competition, the egg breaks if it hits the ground at more than $8\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$ and ignore air resistance.

1. a) Calculate the time of fall corresponding to this speed limit and the maximum height from which the unprotected egg can be dropped.
2. b) If it is dropped from a window at $20\ \text{m}$, with what speed does it arrive? Does it survive?
3. c) Propose two design improvements so that the egg survives a drop from $20\ \text{m}$ and justify them by relating them to the time of fall or the impact speed.