Course 1.1Lesson 5 of 5

Project: the best angle for a throw

A project that uses the whole chapter: a ball is thrown at 20 metres per second at several angles, and arrays give the time of flight, the range and the greatest height for every angle with one formula each. The range is v squared times sin of 2 theta over g, the time of flight is 2 v sin theta over g, and the height is the square of v sin theta over twice g. np.deg2rad changes the angles to radians, argmax finds the angle of the longest range, which is 45 degrees, and a mask finds the angles that give a range above a limit. A column of angles with a row of values of g gives a whole table by broadcasting, and the same angle gives a range six times longer on the Moon than on Earth.

about 30 min1 simulation2 practice questions

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

use a physics formula on a whole array of values,
print a table from several arrays with zip,
find the best value with argmax, and
make a table of results for two quantities with broadcasting.

The problem

A ball is thrown at a speed $v_0 = 20\ \text{m/s}$ at an angle $\theta$ above the ground. If we forget the air, the ball is a projectile. From physics, we know three formulas.

\displaystyle \text{time of flight} = \frac{2 v_0 \sin\theta}{g} \qquad \text{range} = \frac{v_0^2 \sin 2\theta}{g} \qquad \text{greatest height} = \frac{(v_0 \sin\theta)^2}{2g}

Here $g = 9.8\ \text{m/s}^2$ . The questions are: how far does the ball go for each angle, and which angle is best? Before you read on, make a guess. Many people say 45 degrees. Let us see.

The program

Predict first. At which angle will the range be the longest? Then run it.

Read the program.

- angle = np.arange(15, 90, 15) is the five angles in degrees: 15, 30, 45, 60, 75.

- theta = np.deg2rad(angle) changes them to radians, because np.sin needs radians. This is the line that people forget.

- Each of the three formulas becomes one line, and each line works on all five angles at once. This is what you learned in lesson 2.

- zip(angle, time_of_flight, distance, height) goes through the four arrays together, one place at a time, so that the loop can print a row of the table. The f-string {a:>5} puts the number in a width of 5, right-aligned, and {t:5.2f} shows 2 decimals.

- distance.argmax() is the place of the longest range. angle[...] turns that place into the angle: 45 degrees. So the guess was right. Look at the table: 30 and 60 degrees give the same range, 35.3 m, and so do 15 and 75 degrees. The reason is that $\sin 2\theta$ has the same value for $\theta$ and $90 - \theta$ .

- angle[distance > 35] is a mask: the angles for which the range is more than 35 m. They are 30, 45 and 60.

Notice also the last column. The ball reaches a greater height at the steeper angle, but it does not go as far. A high throw stays in the air longer (3.94 s at 75 degrees), but most of its speed is used to go up.

Another planet

The same ball is thrown on Mars or on the Moon. The gravity is different. What happens? Now we have two quantities: the angle and the gravity.

distance.max(axis=0) is the best range of each column, the best for each planet: 40.8 m on Earth, 108.1 m on Mars, and 250.0 m on the Moon. On the Moon, the same throw goes about six times as far, because the gravity is about six times less. The best angle is the same on all of them: 45 degrees.

The hollow dot is the angle 90° − θ. It always sits at the same height as the full dot: the two throws land in the same place.

Checking your result

A good scientist always checks. Three ways to check a result like this one:

A special case. At 90 degrees the ball goes straight up, and the range must be 0. And at 45 degrees, the formula simplifies to $v_0^2 / g = 400 / 9.8 = 40.8$ m, which is the value in the table.
A pattern. The ranges at 30 and 60 degrees are equal, as the physics says.
The units. The range is in metres, the time in seconds. If the range came out as 0.004 or 40 000, you would suspect a mistake such as degrees used where radians are needed.

Common mistakes

Forgetting np.deg2rad. The numbers look strange, and the longest range is not at 45 degrees.
Using math.sin on an array. It gives a TypeError. Use np.sin.
**Writing sin(2 * theta) for sin(theta) ** 2.** Read the formula again.
Reshaping the wrong array for broadcasting. The angle must be a column and the gravity a row.

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: the best angle. You write the functions for the time of flight, the range and the best angle.
Build. Open the problem The best angle on the Moon.
Stretch. Open the problem A throw from a roof, where the ball starts above the ground.

Remember these

A formula for one value works on a whole array of values. Write it once.
Change degrees to radians with np.deg2rad before you use np.sin.
zip goes through several arrays together. argmax gives the place of the biggest number.
A column and a row, combined, make a table by broadcasting.
Check a result: a special case, a pattern, and the units.

Check yourself

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

Why must the angle go through np.deg2rad?

At which angle is the range the longest on Earth? And on the Moon?

Why do 30 and 60 degrees give the same range?

What is the shape of the table of ranges for 4 angles and 5 values of gravity?

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Problems (2)

A ball is thrown at 20 m/s on the Moon, where g = 1.6 m/s².Medium · High School (11–12) · by StemPy Lab

A ball is thrown horizontally at 15 m/s from a roof that is 20 m high.Hard · High School (11–12) · by StemPy Lab

Python Labs (1)

That was the last lessonBack to the course: take the test

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