# Putt Physics Calculation 03: Putter-Ball Collision

In our last article, we talked about calculating the swing range in the golf app “Putt Physics”. By combining it with the putter head speed we calculated in the previous article, we were able to determine the swing range by finding the putter head speed at the lowest point.

Now, in our third article, we finally introduce the ball! We will calculate the motion of the putter head and the ball at the moment of collision, and determine the initial velocity of the golf ball. This calculation uses the principle of conservation of momentum.

The equation is as follows:

$$m_{a}v_{a0}+m_{b}v_{b0}=m_{a}v_{a}+m_{b}v_{b}$$

Here are the meanings of each symbol:

$$m_a$$:　Weight of putter
$$m_b$$:　Weight of the ball
$$v_{a0}$$:　Putter head speed just before collision
$$v_{b0}$$:　Ball speed just before collision
$$v_a$$:　Putter head speed immediately after collision
$$v_b$$:　Ball velocity immediately after collision

The golf ball is hit from a stationary position, so the ball velocity before collision, vb0, is 0. The next thing to consider is the coefficient of restitution, e. This value represents the magnitude of the rebound force generated by the collision between the putter and the ball.

The coefficient of repulsion, e, is expressed by the following equation

$$e=-\frac{v_a-v_b}{v_{a0}-v_{b0}}=-\frac{v_a-v_b}{v_{a0}}$$

In reality, it often takes a value between 0 and 1, but for simplicity, we assume e=1 in this case. Based on this assumption, we can transform the previous equation as follows:

$$v_a=v_b-v_{a0}$$

To calculate the motion at the moment of collision between the golf ball and the putter head, we use the principle of conservation of momentum. By using this equation, we can determine the initial velocity of the golf ball.

Let’s crunch some numbers, shall we? According to the rules of golf, the weight of a golf ball $$m_b$$ is set at 45.93g. But what about the weight of a putter $$m_a$$? Well, it depends on whether you’re just considering the weight of the putter head (which is usually around 350g), the weight of the entire putter club (which can range from 500-600g), or even factoring in the weight of a human body (which could bring the total weight up to around 50,000g). Let’s assume ma could range from a minimum of 350g to a maximum of 50,000g and calculate from there.

If the head of the putter is moving at a speed of 1m/s, the initial speed of the ball $$v_b$$ can be calculated using the following equation:

$$v_b=\frac{2m_{a}v_{a0}}{m_a+m_b}$$

$$m_a=350g, then v_b=\frac{2*350*1}{350+45.93}=1.77$$

$$m_a=50000g, then v_b=\frac{2*50000*1}{50000+45.93}=1.998$$

So what does this mean? Well, it means that no matter how heavy you make your putter, the initial speed of the ball won’t be more than twice the speed of the putter head. But here’s the thing – when I actually go out and putt, sometimes the ball’s initial speed is higher than what the calculations predict. Why is that? Could it have something to do with how force is transmitted?

Let’s dig deeper!

When putting, it’s important to make sure that the putter head doesn’t decelerate after making contact with the ball. Personally, I’ve been taught by more experienced players to make my backswing and follow-through the same length, and I’ve found that this method works well for me. However, the length and rhythm of your follow-through can vary from person to person. Regardless, it’s important to make sure your follow-through doesn’t lose to the ball’s collision. If you don’t pay attention, the ball’s trajectory can end up wonky.

So basically, even if you’re not intentionally making a punch shot, you’re probably still putting in a certain amount of punch-like force. But since this force isn’t reflected in the equation, we might need to make some adjustments.

$$m_{a}v_{a0}+m_{b}v_{b0}+ ??? =m_{a}v_{a}+m_{b}v_{b}$$

But I do think that the proportionality between the head speed $$v_a0$$ and the ball’s initial speed $$v_b$$ still holds true. That’s why we’ve defined the ratio between the two as the putter transfer rate in our app, “Putt Physics.” We set the initial value of this rate to be greater than 2 (specifically 3.5), but it can vary depending on the putter’s characteristics and the golfer’s swing method. That’s why it’s important to adjust your own putter transfer rate as you putt.

So there you have it – the discussion on collision calculations is over.

When it comes to putting, there are many small factors to consider, but ultimately, it’s important to find the method that works best for you and your own unique style. And hey, don’t forget to have fun out there on the greens!

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