Determining the Ratio of Maximum Heights for Two Balls Projected at Different Angles

When dealing with the physics of projectiles, one common question is to determine the ratio of the maximum heights reached by two balls projected at different angles but with the same initial velocity. In this article, we explore the mathematical principles behind this phenomenon using the projectile motion equations and discuss the key factors involved.

Mathematical Formulas and Notations

Let's denote:

- u as the initial velocity,

- θ as the angle of projection, and

- g as the acceleration due to gravity (which is approximately -9.80665 m/s2).

Projectile Motion Equation

The formula for the maximum height H of a projectile can be derived from the vertical motion equation:

1/2 mu 2sin2θ mg H

Thus, the maximum height H can be given by:

H u2sin2θ / (2 g)

Calculations for Different Angles

Let's consider two balls projected at angles of 30° and 60° with the same initial velocity u.

For the 30° Angle

The maximum height H30 can be calculated as:

H30 u2sin230° / (2 g)

Since sin(30°) 0.5, the expression simplifies to:

H30 u2 * (0.5)2 / (2 g) u2 * 0.25 / (2 g) u2 / (8 g)

For the 60° Angle

The maximum height H60 can be calculated similarly:

H60 u2sin260° / (2 g)

Since sin(60°) √3/2 ≈ 0.866, the expression simplifies to:

H60 u2 * (0.866)2 / (2 g) u2 * 0.75 / (2 g) 3u2 / (8 g)

Ratio of Maximum Heights

To find the ratio of the maximum heights, we divide H60 by H30 as follows:

Ratio H60 / H30 (3u2 / (8 g)) / (u2 / (8 g)) 3

Conclusion

The ratio of the maximum heights of the two balls is 3:1, indicating that the ball projected at 60° will reach three times the maximum height as the ball projected at 30°, given the same initial velocity.

Additional Resources and Links

For more detailed information on projectile motion and related topics, you can refer to the following resources:

Projectile motion on PhysicsClassroom Projectile motion on Wikipedia