Free fall
Stand at the top of the Cliffs of Moher and drop a pebble: how long does it take to reach the Atlantic, and how fast is it moving when it gets there? You will answer this in the next section. On the Moon, where there is no air, a hammer and a feather dropped at the same time reach the ground together. On Earth they fall differently only because of air resistance, but the cause of the motion (gravity) is the same for both. Free fall is motion in a straight vertical line with a constant acceleration that does not depend on the mass of the object.
Gravity as a constant acceleration
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…
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.
- a) What does this fact show about falling objects when there is no air resistance?
- 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. a) Calculate its speed at $1\ \text{s}$, $2\ \text{s}$…
In free fall, the speed of an object increases by $9.8\ \text{m/s}$ every second. An object is dropped from rest.
- a) Calculate its speed at $1\ \text{s}$, $2\ \text{s}$ and $3\ \text{s}$.
- b) What acceleration does the object have during the fall, and which letter is used to represent it?
Speed and distance in a fall
When you measure the fall of a ball from rest, you observe that in $1\ \text{s}$ it covers about $5\ \text{m}$, in $2\ \text{s}$ it has already covered $20\ \text{m}$ and in $3\ \text{s}$ it reaches $45\ \text{m}$: the distance does not increase at the same rate, but shoots up with each second. What mathematical relationship links time with the distance covered?
Look at the velocity–time graph: it is a straight line that always rises with the same gradient, $g$. The displacement–time graph, $s(t)$ (labelled $x$ in the widget), curves upwards faster and faster, because the velocity keeps increasing.
To find the distance fallen during one particular second, work out $s$ at the end of that second and at the start of it, and subtract. To change $\text{m s}^{-1}$ into $\text{km h}^{-1}$, multiply by $3.6$. If a distance is given in cm, change it to metres before using the equations.
A stone is dropped from rest. With $g = 9.8\ \text{m/s}^2$, calculate its velocity after $2\ \text{s}$ of falling using $v = v_0 + g \times t$.
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 $h = \frac{1}{2} g \times t^2$.
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 $h = \frac{1}{2} g \times t^2$.
An object is dropped from a bridge $80\ \text{m}$ high. Take $g \approx 10\ \text{m/s}^2$. a) Calculate the time of fall. b) Calculate the velocity upon reaching the…
An object is dropped from a bridge $80\ \text{m}$ high. Take $g \approx 10\ \text{m/s}^2$.
- a) Calculate the time of fall.
- b) Calculate the velocity upon reaching the water, in $\text{m/s}$.
- c) Express this velocity in $\text{km/h}$ and comment on why jumping from a bridge is dangerous even into water.
Vertical projection upwards
A ball is thrown vertically upwards at $15\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. a) How long does it take to reach its maximum height? Hint: at the highest…
A ball is thrown vertically upwards at $15\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.
- a) How long does it take to reach its maximum height? Hint: at the highest point, the velocity is zero.
- 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 $g \approx 10\ \text{m/s}^2$. a) What maximum height does it reach above the level of the…
A stone is thrown vertically upwards from a bridge at $10\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.
- a) What maximum height does it reach above the level of the bridge?
- b) How long does it take to pass the level of the bridge again?
Symmetry of rising and falling
If you toss a sliotar straight up and time its flight, you will find that it takes almost exactly the same time to rise until it stops as to return to your hand. Why do the rise and the fall look like two identical halves of the same film played forwards and backwards?
You throw a ball vertically upwards at $20\ \text{m/s}$ from the ground, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$. a) What maximum height does it reach?…
You throw a ball vertically upwards at $20\ \text{m/s}$ from the ground, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.
- a) What maximum height does it reach?
- b) How much time passes until it returns to your hands?
- 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 $g \approx 10\ \text{m/s}^2$. a) Calculate the maximum height and…
A ball is thrown vertically upwards at $25\ \text{m/s}$ from the ground, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.
- a) Calculate the maximum height and the time at which it reaches it.
- 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.
- c) Calculate the velocity of the ball for each of these two solutions.
- d) Explain the physical meaning of the two solutions and the relationship between the two velocities.
Combining free fall and vertical projection: two objects
An object released from a moving carrier, such as a bag dropped from a rising hot air balloon, keeps the velocity the carrier had at that instant (Newton's first law). So $u$ is the velocity of the carrier, and the object keeps rising for a moment before gravity brings it to rest.
Two balls are dropped from a height of $45\ \text{m}$, the second one second after the first. Take $g \approx 10\ \text{m/s}^2$. a) How long does the first ball take to…
Two balls are dropped from a height of $45\ \text{m}$, the second one second after the first. Take $g \approx 10\ \text{m/s}^2$.
- a) How long does the first ball take to reach the ground?
- 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?
- 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.
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 $40\ \text{m/s}$,…
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 $g \approx 10\ \text{m/s}^2$.
- a) Write the height equations for each stone, $h_1$ and $h_2$, measured from the ground.
- b) Calculate the instant at which they cross paths.
- c) At what height do they cross paths?
- 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…
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 $g \approx 10\ \text{m/s}^2$.
- a) Explain why the bag, upon coming loose, continues to move upwards for a moment instead of falling immediately.
- 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.
- c) With what velocity does it hit the ground?
Measuring g in the lab
How do we know that $g = 9.8\ \text{m s}^{-2}$? You can measure it. Release a steel ball from an electromagnet at a height $s$ above the floor and time the fall with a light gate and a data logger. Measure $s$ with a metre stick. Repeat each height three times and use the average time to reduce random error.
From $s = \frac{1}{2}gt^2$, a graph of $s$ against $t^2$ is a straight line through the origin with gradient $\frac{g}{2}$. This is how you verify the model of free fall with primary data: if your points lie close to a straight line through the origin, the model $s = \frac{1}{2}gt^2$ (constant acceleration) holds, and the gradient then gives you $g$.
The same idea works with a ruler: if a friend drops it and you catch it after it has fallen $s$ (in metres), your reaction time is $t = \sqrt{2s/g}$.
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 $20\ \text{cm}$,…
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 $g \approx 10\ \text{m/s}^2$.
- a) Express the distance in metres and calculate the time the ruler has been falling.
- 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?
A diver drops from a diving board at a height of $19.6\ \text{m}$. a) With $g = 9.8\ \text{m/s}^2$, calculate the time of fall and the speed of entry into the water. b)…
A diver drops from a diving board at a height of $19.6\ \text{m}$.
- a) With $g = 9.8\ \text{m/s}^2$, calculate the time of fall and the speed of entry into the water.
- b) Repeat the calculation using the approximation $g \approx 10\ \text{m/s}^2$, rounding to two decimal places.
- 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.