Here is a question that was asked on my Probability and Stats for Engineers exam - Let's see how you do::
You are challenged with designing the geometry of the face of a golf club using probabilistic considerations. You will first have to derive a formula for the flight distance achieved (under some assumptions) by a golf ball struck by this golf club, and then use probabilistic concepts to choose a golf club face design (face angle and coefficient of restitution, along with specifying accuracy of swing speed and angle) that meets some given specifications on accuracy and limits on variations in certain quantities.
The golf club to be designed is used for full-swing situations with unimpeded contact of the club face and the ball - typical of a driver or fairway wood for those familiar with golf. We assume that a typical golfer generates a full swing club speed for such a shot that varies randomly with a mean of 90 mph and a standard deviation of 3 mph. [FYI: Pro golfers, men and women, typically achieve club head speeds of 110 mph or more with very little variation. These guys are good!] The impact of club head and ball is then essentially a collision between an object with a specified direction and speed (the club head) and a stationary object (the ball). We will further assume that the coefficient of restitution e between the golf ball and the club face is something less than 1 (around 0.78 is a typical assumption - the USGA actually restricts this value to less than 0.85).
In this project, you must choose this coefficient of restitution and also specify its tolerable random variation, among other things. (See Mechanics II for discussion of coefficient of restitution in collisions.) You can assume that the golf ball’s mass is 1/6 of the club head’s mass. Golf balls are about 40 grams, while driver and wood club heads are typically 200 grams or more. It turns out that more club head mass makes very little difference to distance, so making the club head heavy as a brick really does not help much! Remember this when a golf club salesperson wants you to buy the Enormous Giganto Big Boomer Solid Brick driver.
For computing the ball’s flight trajectory and distance, we will ignore air drag, ignore air
currents, and ignore the effect of the ball spinning as it flies - in other words, we will neglect all of the aerodynamics, just like we always do in Aerospace Engineering!! You should consider only where the ball lands initially, not where it bounces or rolls to after first striking the ground (which typically adds another 20% to the distance). You can assume that the landing area is at the same altitude or height as the point where the ball is struck (a nice flat Midwestern golf hole). Our golfer will launch the ball with an initial trajectory angle that is determined by the club face angle and, of course, the actual trajectory of the club head at impact. If the club head path is exactly horizontal at impact, the initial trajectory of the ball will be at an angle that is the same as the club face angle - for this type of golf club, typically about 15 degrees above horizontal. However, this angle will be slightly in error due to our golfer’s swing trajectory being a bit off horizontal. Of course, the initial flight angle for maximum ballistic flight distance is actually 45 degrees above horizontal (do you know why?). But if the club face angle is anywhere near that large, then much of the club head speed at impact is transformed into vertical ball velocity, and the initial horizontal ball velocity is reduced, resulting in little extra distance from bouncing or rolling - great for iron shots where you want the ball to stop right where it lands, not so good when distance is the objective. When aerodynamic effects (especially of spinning) and the rolling and bouncing of the ball after its initial ground strike are accounted for, the total distance covered by the ball is substantially larger when the initial flight angle (and hence the club face angle) is much shallower than 45 degrees. [FYI: Pro golfers use drivers with club face angles of less than 10 deg and hit the ball off the tee at an initial flight angle slightly above the club face angle - usually around 13 deg - with a lot of backspin that generates lift as the ball flies. That is how they hit those 300-yard drives.]
Your design mission is to:
Choose the coefficient of restitution e and its tolerable error,
Choose the tolerable random error in the initial angle of the trajectory, and
Modify the given error (3 mph) in the club head velocity at impact, if necessary
such that:
1. The distance to the initial landing site of the ball is accurate to within 12 yards (36 feet) in either direction (short or long) with 95.5% confidence, and: 2. You have 99% confidence you have not violated the USGA rule that e < 0.85 (notice that this a “one-sided” condition, you don’t care if the actual e is considerably below your chosen e).
You will have to determine a reasonable allocation of the tolerable error to e, to the initial angle of impact, and to the club head velocity at impact, and there is not a unique way to do this. Furthermore, any reduction that you suggest to the given club head speed uncertainty should be as small as possible because average golfers’ swings are difficult to make repeatable! Your results should include the mean flight distance and the 95.5% confidence band around this distance (which obviously must be no larger than the specification of +/-12 yds) for your design. The flight distance of the ball will be (trust me!) a nasty nonlinear function of the coefficient of restitution, the club head speed on impact, and the initial flight angle(= club face angle when no swing error occurs), which you will have to derive. [Hints on derivation: Because the club face is at an angle, the collision is oblique even when the club head is traveling exactly horizontally at impact. As a result, right after the collision, the ball has upward and forward components of velocity, while the club head has forward and downward components of velocity. In practice, the golfer holds on to the club through the swing and therefore quickly cancels the downward
You are challenged with designing the geometry of the face of a golf club using probabilistic considerations. You will first have to derive a formula for the flight distance achieved (under some assumptions) by a golf ball struck by this golf club, and then use probabilistic concepts to choose a golf club face design (face angle and coefficient of restitution, along with specifying accuracy of swing speed and angle) that meets some given specifications on accuracy and limits on variations in certain quantities.
The golf club to be designed is used for full-swing situations with unimpeded contact of the club face and the ball - typical of a driver or fairway wood for those familiar with golf. We assume that a typical golfer generates a full swing club speed for such a shot that varies randomly with a mean of 90 mph and a standard deviation of 3 mph. [FYI: Pro golfers, men and women, typically achieve club head speeds of 110 mph or more with very little variation. These guys are good!] The impact of club head and ball is then essentially a collision between an object with a specified direction and speed (the club head) and a stationary object (the ball). We will further assume that the coefficient of restitution e between the golf ball and the club face is something less than 1 (around 0.78 is a typical assumption - the USGA actually restricts this value to less than 0.85).
In this project, you must choose this coefficient of restitution and also specify its tolerable random variation, among other things. (See Mechanics II for discussion of coefficient of restitution in collisions.) You can assume that the golf ball’s mass is 1/6 of the club head’s mass. Golf balls are about 40 grams, while driver and wood club heads are typically 200 grams or more. It turns out that more club head mass makes very little difference to distance, so making the club head heavy as a brick really does not help much! Remember this when a golf club salesperson wants you to buy the Enormous Giganto Big Boomer Solid Brick driver.
For computing the ball’s flight trajectory and distance, we will ignore air drag, ignore air
currents, and ignore the effect of the ball spinning as it flies - in other words, we will neglect all of the aerodynamics, just like we always do in Aerospace Engineering!! You should consider only where the ball lands initially, not where it bounces or rolls to after first striking the ground (which typically adds another 20% to the distance). You can assume that the landing area is at the same altitude or height as the point where the ball is struck (a nice flat Midwestern golf hole). Our golfer will launch the ball with an initial trajectory angle that is determined by the club face angle and, of course, the actual trajectory of the club head at impact. If the club head path is exactly horizontal at impact, the initial trajectory of the ball will be at an angle that is the same as the club face angle - for this type of golf club, typically about 15 degrees above horizontal. However, this angle will be slightly in error due to our golfer’s swing trajectory being a bit off horizontal. Of course, the initial flight angle for maximum ballistic flight distance is actually 45 degrees above horizontal (do you know why?). But if the club face angle is anywhere near that large, then much of the club head speed at impact is transformed into vertical ball velocity, and the initial horizontal ball velocity is reduced, resulting in little extra distance from bouncing or rolling - great for iron shots where you want the ball to stop right where it lands, not so good when distance is the objective. When aerodynamic effects (especially of spinning) and the rolling and bouncing of the ball after its initial ground strike are accounted for, the total distance covered by the ball is substantially larger when the initial flight angle (and hence the club face angle) is much shallower than 45 degrees. [FYI: Pro golfers use drivers with club face angles of less than 10 deg and hit the ball off the tee at an initial flight angle slightly above the club face angle - usually around 13 deg - with a lot of backspin that generates lift as the ball flies. That is how they hit those 300-yard drives.]
Your design mission is to:
Choose the coefficient of restitution e and its tolerable error,
Choose the tolerable random error in the initial angle of the trajectory, and
Modify the given error (3 mph) in the club head velocity at impact, if necessary
such that:
1. The distance to the initial landing site of the ball is accurate to within 12 yards (36 feet) in either direction (short or long) with 95.5% confidence, and: 2. You have 99% confidence you have not violated the USGA rule that e < 0.85 (notice that this a “one-sided” condition, you don’t care if the actual e is considerably below your chosen e).
You will have to determine a reasonable allocation of the tolerable error to e, to the initial angle of impact, and to the club head velocity at impact, and there is not a unique way to do this. Furthermore, any reduction that you suggest to the given club head speed uncertainty should be as small as possible because average golfers’ swings are difficult to make repeatable! Your results should include the mean flight distance and the 95.5% confidence band around this distance (which obviously must be no larger than the specification of +/-12 yds) for your design. The flight distance of the ball will be (trust me!) a nasty nonlinear function of the coefficient of restitution, the club head speed on impact, and the initial flight angle(= club face angle when no swing error occurs), which you will have to derive. [Hints on derivation: Because the club face is at an angle, the collision is oblique even when the club head is traveling exactly horizontally at impact. As a result, right after the collision, the ball has upward and forward components of velocity, while the club head has forward and downward components of velocity. In practice, the golfer holds on to the club through the swing and therefore quickly cancels the downward
