How to Calculate the Height of a Thrown Ball

TL;DR

To calculate the height a ball reaches when thrown vertically, use the equation h = -gt²/8, where g is the acceleration due to gravity and t is the total time of flight. This formula shows that greater time in the air results in higher altitude, and the average velocity during this time will be zero as the ball returns to its starting point.

Transcript

Welcome back. Now I think that hopefully you have some intuition, and maybe you can figure this out on your own, but I'll show how to do it. We spent a lot of time on how fast did I throw the ball-- how fast was it when it left my hand-- so let's figure out how high did it go, because that's kind of a cool thing to know, as well. We figured out tha... Read More

Key Insights

• 🏐 The initial velocity of a thrown ball can be found using the equation v = -gt/2.
• 💬 The height a ball reaches when thrown vertically can be calculated using the equation h = -gt^2/8.
• 💦 The same equations can be used to calculate the time it takes for a dropped ball to fall.
• ❤️‍🩹 Average velocity depends on both speed and direction, and in this case, it is 0 because the ball starts and ends at the same place.
• 🐎 Understanding the concepts of speed and velocity is crucial in differentiating between them.
• 👱 The equations used in this analysis rely on the assumptions of idealized motion without air resistance or other external factors.
• 💬 Throwing a ball with a longer time of flight results in a greater height achieved.

Questions & Answers

Q: How do you calculate the initial velocity of a thrown ball?

The initial velocity can be calculated using the equation v = -gt/2, where g is the acceleration due to gravity and t is the time. For example, if the time is 3 seconds, the initial velocity would be -1.5g m/s.

Q: What is the formula for calculating the height a ball reaches when thrown vertically?

The height can be calculated using the equation h = -gt^2/8, or alternatively, h = -g/2 * (t/2)^2. For instance, if the time is 4 seconds, the height would be -2g m.

Q: Can the same equations be used for a dropped ball?

Yes, the equations can be used for a dropped ball as well. If the ball is dropped from a height h, we can use the equation -h = -g/2 * (t/2)^2 to calculate the time it takes for the ball to fall.

Q: How do speed and velocity differ in this context?

Speed represents the magnitude of the velocity, while velocity also includes the direction. In the video, the average velocity over the entire time period is 0 because the ball starts and ends at the same place, despite having varying speeds during its trajectory.

Summary & Key Takeaways

• The initial velocity of a thrown ball can be calculated using the equation v = -gt/2, where g is the acceleration due to gravity and t is the time.

• The height the ball reaches can be found using the equation h = -gt^2/8, or alternatively, h = -g/2 * (t/2)^2.

• The same equations can be used to calculate the time it takes for a dropped ball to fall from a given height.