Question

an olympic athlete throws a javelin at four different angles above the horizontal, each with the same speed: 30°, 40°, 60°, and 80°. which two throws cause the javelin to land the same distance away?

Answer 1

The two throws that cause the javelin to land the same distance away are:

1. The throws at 30° and 60°.

2. The throws at 40° and 50°.

To determine which two throws cause the javelin to land the same distance away, we need to consider the range of a projectile. The range
`(\(R\))` of a projectile launched at an angle
`\(\theta\)` with an initial speed
`\(v_0\)` can be calculated using the formula:

`\[R = (v_0^2 \sin(2\theta))/(g),\]`

where
`\(g\)` is the acceleration due to gravity.

In this case, the athlete throws the javelin at four different angles with the same speed. Let's denote the initial speed as
`\(v_0\)` (which is the same for all throws). We can compare the ranges for different angles.

1. For an angle of 30°
`(\(\theta = 30^\circ\))`:

`\[R_(30) = (v_0^2 \sin(2 * 30^\circ))/(g).\]`

2. For an angle of 40°
`(\(\theta = 40^\circ\))`:

`\[R_(40) = (v_0^2 \sin(2 * 40^\circ))/(g).\]`

3. For an angle of 60°
`(\(\theta = 60^\circ\)):`

`\[R_(60) = (v_0^2 \sin(2 * 60^\circ))/(g).\]`

4. For an angle of 80°
`(\(\theta = 80^\circ\))`:

`\[R_(80) = (v_0^2 \sin(2 * 80^\circ))/(g).\]`

Now, we can compare these ranges and identify pairs that are equal.

Without specific numerical values, we can make some observations:

- The ranges for angles
`\(30^\circ\) and \(60^\circ\)` are equal because
`\(\sin(2 * 30^\circ) = \sin(2 * 60^\circ)\)`.

- The ranges for angles
`\(40^\circ\) and \(50^\circ\)` are also equal because
`\(\sin(2 * 40^\circ) = \sin(2 * 50^\circ)\)`.