Questions: If you double the throw angle, will the distance the ball travels double? Explain.

Solution Steps

We need to determine if doubling the throw angle of a projectile will result in doubling the distance it travels. This involves understanding the relationship between the throw angle and the distance traveled by a projectile.

The distance \( R \) traveled by a projectile launched with an initial velocity \( v_0 \) at an angle \( \theta \) is given by the formula: \[ R = \frac{v_0^2 \sin(2\theta)}{g} \] where \( g \) is the acceleration due to gravity.

If we double the throw angle, the new angle becomes \( 2\theta \). Substituting \( 2\theta \) into the formula, we get: \[ R' = \frac{v_0^2 \sin(2 \cdot 2\theta)}{g} = \frac{v_0^2 \sin(4\theta)}{g} \]

To compare the original distance \( R \) and the new distance \( R' \), we need to analyze the sine function:

• Original distance: \( R = \frac{v_0^2 \sin(2\theta)}{g} \)
• New distance: \( R' = \frac{v_0^2 \sin(4\theta)}{g} \)

The sine function does not have a linear relationship with its argument. Therefore, \( \sin(4\theta) \) is not simply twice \( \sin(2\theta) \).

Since \( \sin(4\theta) \) is not equal to \( 2 \sin(2\theta) \), doubling the throw angle does not result in doubling the distance traveled by the projectile.

Final Answer