Free fall
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 because of the air, but the cause of the motion is the same: gravity. Free fall is uniformly accelerated motion (UAM) in the vertical direction, with the same acceleration for any object.
Gravity as a constant acceleration
If you drop a coin and a sheet of paper at the same time, the coin falls much faster; but if you scrunch that same sheet into a tight ball, both fall together: does the speed of the fall depend on the mass of the object or on air resistance?
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?
Velocity and distance in free fall
When you take a stroboscopic photograph of a falling ball, the positions recorded at equal time intervals are further and further apart: at exactly what rate do the velocity and the distance covered increase during each second of the fall?
Look at the $v(t)$ graph: it is a straight line that always rises with the same gradient, $g$. The $x(t)$ graph curves more and more steeply.
A stone is dropped from rest. With $g = 9,8\ \text{m/s}^2$, calculate its velocity after $2\ \text{s}$ of falling using $v = v_0 + g \cdot t$.
A stone is dropped from rest. With $g = 9,8\ \text{m/s}^2$, calculate its velocity after $2\ \text{s}$ of falling using $v = v_0 + g \cdot t$.
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 $h = \frac{1}{2} g \cdot t^2$.
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 $h = \frac{1}{2} g \cdot t^2$.
The vertical throw upwards
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.
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.
Combining free fall and vertical throw
If you drop a ball from a balcony at the very moment someone throws another one upwards from the street, the two will cross somewhere in the air: at what height and at exactly what second do they meet?
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…
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 $g \approx 10\ \text{m/s}^2$.
- a) What maximum height does it reach above the roof? And above the ground?
- b) How long does it take to go up?
- 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.
- 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 $40\ \text{m/s}$,…
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.