Course 2.2Lesson 5 of 5

Project: a ball with air resistance

A project that uses the whole chapter: a ball is thrown at 40 metres per second at 45 degrees, and air resistance makes the force on it proportional to the square of its speed. The state is the position and the velocity, four numbers, and solve_ivp follows them with a rate function. An event function that returns the height, with terminal true and direction minus 1, stops the solution when the ball reaches the ground. Without air the range is 163.3 metres, equal to v squared over g, and with the drag constant 0.0089 per metre the range is only 82.5 metres, the flight is shorter and the height is lower. A scan of the angles shows that the best angle without air is 45 degrees and with air it is lower, 40 degrees.

about 35 min1 simulation

Lesson 5 of 5Show all lessons

Try the code and the simulations beside the lesson.

What you will be able to do

After this lesson you can

write a model with several quantities as a rate function for solve_ivp,
stop a solution when something happens, with an event, and
use the model to answer a question that has no formula: the best angle with air resistance.

The problem

In chapter 1, you found the best angle for a throw: 45 degrees, and the range $v_0^2 / g$ . But that was for a ball in empty space. A real ball moves in air, which slows it down. A cricket ball, a football and an arrow all feel the drag, and for quick objects it is not small. What are the range, the time and the height, and is 45 degrees still the best angle?

There is no simple formula. For this kind of problem, the computer is the tool.

The model

The force of the air on a body in fast motion is proportional to the square of the speed, and it points against the velocity. As an acceleration:

\displaystyle \vec{a}_{air} = -k\,|\vec{v}|\,\vec{v}

where $|\vec{v}|$ is the speed and $k$ is a constant that depends on the shape, the size and the mass: $k = \dfrac{\rho C_d A}{2m}$ , where $\rho = 1.2$ kg/m $^3$ is the density of air, $C_d$ is the drag coefficient, $A$ is the area, and $m$ the mass. For a ball with $C_d = 0.5$ , $A = 0.0043$ m $^2$ and $m = 0.145$ kg (a baseball), $k = 0.0089$ per metre. Gravity adds the acceleration $-g$ in the vertical direction.

The motion has four quantities: the position $(x, y)$ and the velocity $(v_x, v_y)$ . Their rates of change are:

$\dfrac{dx}{dt} = v_x$ and $\dfrac{dy}{dt} = v_y$ ,
$\dfrac{dv_x}{dt} = -k\,v\,v_x$ and $\dfrac{dv_y}{dt} = -g - k\,v\,v_y$ , with $v = \sqrt{v_x^2 + v_y^2}$ .

Notice that the equations for the $x$ and $y$ directions are coupled: the speed $v$ contains both $v_x$ and $v_y$ . This is why there is no simple formula.

The program

Predict first. Without air, the range at 40 m/s and 45 degrees is $40^2 / 9.8 = 163$ m. What do you expect with the air? Half? A tenth? Then run it.

Read the program.

- flight(v0, angle, k) is a function that does the whole job for one throw. The angle is changed to radians.

- rate(t, s) unpacks the state s = [x, y, vx, vy], computes the speed with np.hypot(vx, vy) (the square root of $v_x^2 + v_y^2$ ), and returns the four rates of change.

- The event. How long should the solution run? Until the ball lands, but we do not know when. landed(t, s) returns the height s[1], and solve_ivp watches this function: when it changes sign, an event happens. landed.terminal = True says "stop the solution at this event", and landed.direction = -1 says "only when the height is going from positive to negative" (so that the start, when the height is 0 and the ball is going up, does not stop it).

- (0, 100) is a long time span. The event stops the solution at the landing, so the 100 is not reached.

- dense_output=True makes sol.sol(t) a function that gives the state at any time, so that the greatest height can be found by looking at 400 times.

- The results: sol.y[0, -1] is the $x$ at the last time (the range), and sol.t[-1] is the flight time.

The output:

With $k = 0$ : range 163.3 m, time 5.77 s, height 40.8 m. These agree with the formula, $v_0^2/g = 163.3$ m, which is a good check that the program is right.
With $k = 0.0089$ : range 82.5 m, time 4.67 s, height 27.0 m. The air halves the range.

The best angle

Predict first. With air, will the best angle be larger than 45 degrees, equal to it, or smaller?

Why is the best angle smaller with air? A ball thrown at a high angle spends a long time in the air, going up and coming down, and the air slows it for all that time. A flatter throw stays in the air for a shorter time, and loses less speed. So it is better to throw a little lower. This is what athletes know: a javelin or a long jump has a take-off angle that is below 45 degrees. The difference in range between 40 and 45 degrees is only about 1 m, so the best angle is not sharp, but the computer found a real physical effect that the simple formula could not.

The paths

A graph titled A ball thrown at 40 metres per second and 45 degrees. A wide blue arch, with no air, reaches about 41 metres high and lands about 163 metres away. A lower orange arch, with air drag, reaches about 27 metres and lands about 82 metres away, and its fall is steeper than its rise.

The small difference between the program and the formula comes from the size of dt. Make dt smaller in the code and it shrinks.

How to trust the result

A numerical model should always be checked. We did three checks:

A special case. With $k = 0$ , the program gives the textbook result.
A change in the right direction. With air, the range, the time and the height all fall. That is what we expect.
A change of the accuracy. The results should not change when we make rtol and atol smaller. (Try it.)

Also ask: are the numbers sensible? A baseball hit at 40 m/s (about 144 km/h) does travel about 80 to 100 m in air. The result is in the right range.

Common mistakes

Forgetting that the state is a list of four numbers and returning only two rates.
Forgetting the event direction. The solution stops at the start.
Giving the angle in degrees to np.sin and np.cos. Change it to radians.
A span that is too short. The ball has not yet landed when the solution ends.
Believing the model without a check with $k = 0$ .

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

Warm-up. Open the lab Project: landing range. You write the function flight that gives the range, the time and the height.
Build. Open the problem A heavy ball. A heavier ball has a smaller $k$ .
Stretch. Open the problem Does the best angle depend on the speed?

Remember these

A model with several quantities is a rate function that takes the state s and returns a list of the rates.
An event function with terminal = True and a direction stops the solution when something happens, such as a landing.
Air drag lowers the best angle below 45 degrees and halves the range of a fast ball.
Check a numerical model with a special case that you can solve by hand.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

How many quantities are in the state of the ball with air drag in a plane?

What does landed.direction = -1 mean?

Why does k = 0 make a good check?

Why is the best angle lower with air?

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Problems (2)

The drag constant is k = (ρ Cd A)/2m, so for a ball of the same size and shape, it is inversely…Medium · High School (11–12) · by StemPy Lab

With the drag constant k = 0.0089 per metre, find the best angle (whole degrees, from 20 to 70) for…Hard · High School (11–12) · by StemPy Lab

Python Labs (1)

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