Problem: How far can you hit a golf ball on the Moon?
Ok, that was intentionally vague. I’ll make it more precise shortly. But first, I took a slightly roundabout path in getting to this problem. Last week, I read an interesting paper on “The Optimal Angle of Release in Shot Put.” The trajectory of a shot (the heavy metal ball) can be modeled reasonably well assuming a uniform gravitational field and ignoring air resistance… exactly the “nice” situation used in introductory physics textbooks. In that simple case, it is a common exercise to show that, for a given initial speed, the maximum range of a projectile launched from the ground is achieved with a launch angle of 45 degrees.
However, the best shot putters in the world release the shot at an angle of only 37 to 38 degrees. Why is this angle so low? The paper describes the necessary additional assumptions that yield a mathematical solution that agrees with observed practice. I think this is a nice example of the need to be careful about which simplifying assumptions are “safe,” such as neglecting air resistance, and which are not, such as the need to consider the height of release– although this is not the only important additional factor, as the paper shows.
From there, let’s move on to a blog post by Ethan Siegel from a couple of years ago, titled “Could you really hit a golf ball ‘miles and miles’ on the Moon?” (The “miles and miles” is a reference to Alan Shepard’s comment after hitting a golf ball on the Moon during the Apollo 14 mission.) Here on Earth, even the best golfers hit the ball only a few hundred yards. On the Moon, the force of gravity is weaker, and there is no air resistance, so we should be able to hit the ball much farther. The question is, how much farther?
So now let’s make the problem more precise. As in the blog post, our golfer can swing the club so that the club face strikes the ball at 40 meters per second (about 90 miles per hour). Also, we are allowed the same simplifying assumptions:
- The Moon is a flat plane with a uniform gravitational acceleration of 1.624 m/s^2.
- The club-ball collision is perfectly elastic, and the mass of the ball is negligible compared to the mass of the club.
The problem: at what angle should the golfer hit the ball (i.e., what is the loft angle of the club) to maximize the distance the ball travels? Hint: the answer is not 45 degrees as suggested in the blog post, and accordingly, the maximum distance is significantly less than “nearly two and a half miles.”