Range, height and the angle of projection

The farthest throw

You throw a ball as hard as you can, always at the same speed. Which angle sends it farthest? Too low, and it hits the ground quickly. Too high, and it goes up a lot but not far forward. The best angle is somewhere in between.

The range, angle by angle

From the last lesson, on level ground,

The largest value of $\sin 2\theta$ is $1$, when $2\theta = 90^\circ$. So:

• The range is largest at $\theta = 45^\circ$.
• That largest range is $R_{\max} = \dfrac{u^2}{g}$.

For example, a ball thrown at $30\ \text{m/s}$ can go at most $\dfrac{(30\ \text{m/s})^2}{9.8\ \text{m/s}^2} = 91.8\ \text{m}$.

Two angles, one range

Look at the picture. The $30^\circ$ and $60^\circ$ paths land at the same spot. So do the $15^\circ$ and $75^\circ$ paths. Why?

Because $\sin 2\theta = \sin(180^\circ - 2\theta)$. So the angles $\theta$ and $90^\circ - \theta$ give the same range. They are called complementary angles.

The two paths are very different, though. Try it.

For a launch at $25\ \text{m/s}$:

The steeper throw goes much higher and stays up more than twice as long. A goalkeeper's long kick uses the low path to reach a team-mate quickly. A high, slow kick gives the team time to run under the ball.

Finding the angle from the height and the range

Divide the greatest height by the range:

So

The launch speed cancels out. If you know how high a ball went and how far, you know the angle it was thrown at.

Worked example. A ball rises $15\ \text{m}$ and lands $52\ \text{m}$ away. Then $\tan\theta = \dfrac{4 \times 15\ \text{m}}{52\ \text{m}} = 1.154$, so $\theta = 49.1^\circ$.

A special case. At $45^\circ$, $\tan\theta = 1$, so $R = 4H$. The best throw goes four times as far as it goes high.

Speed at a given height

Using $v_y^2 = u_y^2 - 2gy$, and the fact that $v_x$ never changes:

The speed at a height $y$ depends only on that height, not on the angle. Two stones thrown at the same speed at different angles have the same speed whenever they are at the same height. The energy idea in a later chapter says the same thing.

Landing above or below the start

Real throws often land at another level: a ball thrown from a balcony, or onto a roof. Then $T = \dfrac{2u\sin\theta}{g}$ is wrong. Go back to the full equation for $y$, with the landing height put in.

Worked example. A ball is thrown at $15\ \text{m/s}$, at $30^\circ$ above the horizontal, from a balcony $10\ \text{m}$ above the ground. When and where does it land?

1. Take the ground as $y = 0$, so the ball starts at $y = 10\ \text{m}$. It lands when $0 = 10\ \text{m} + 7.5\ \text{m/s} \times t - 4.9\ \text{m/s}^2 \times t^2$.
2. Solve the quadratic. Here $7.5^2 + 4 \times 4.9 \times 10 = 252.25$, so $t = \dfrac{7.5 + \sqrt{252.25}}{9.8} = \dfrac{7.5 + 15.88}{9.8} = 2.39\ \text{s}$. (The other root is negative: a time before the throw.)
3. Range: $x = 15\ \text{m/s} \times \cos 30^\circ \times 2.39\ \text{s} = 31.0\ \text{m}$.

The simulation in the last lesson has a launch height slider. Try it: from a height, the best angle is less than $45^\circ$.

In real life, the air also matters. It slows the ball, so the best angle for a real throw is usually a little less than $45^\circ$. Long jumpers take off at about $20^\circ$, because jumping steeply would cost them too much running speed.

Practise with new numbers

Try it in Python

The program prints the range and the greatest height for launch angles from $15^\circ$ to $75^\circ$, all at $25\ \text{m/s}$.

Remember these

• $R = \dfrac{u^2\sin 2\theta}{g}$ is largest at $45^\circ$, where $R_{\max} = \dfrac{u^2}{g}$.
• $\theta$ and $90^\circ - \theta$ give the same range. The steeper path is higher and slower.
• $\tan\theta = \dfrac{4H}{R}$, and at $45^\circ$, $R = 4H$.
• If the landing is at another level, solve the equation for $y$ with that level.

Check yourself

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

1. What is the greatest range of a ball thrown at $30\ \text{m/s}$?

2. Which other launch angle gives the same range as $20^\circ$?

3. A ball rises $12\ \text{m}$ and lands $24\ \text{m}$ away on level ground. At what angle was it thrown?

4. Two balls are thrown at the same speed, at $35^\circ$ and at $55^\circ$. Which lands first?