In the previous article, we calculated the rolling of a ball on a horizontal plane, so this time we will calculate the motion of a ball rolling on a slope. The formulas are more complicated because of the change from a plane to a slope, but the physical phenomena remain the same. Since the horizontal plane is now a slope, the force diagram will look like this.

The angle of the slope is θ°. In this case, the force mg applied to the ball remains the same as on a horizontal surface. However, due to the angle, it is necessary to separate the force into two directions: one parallel to the slope and the other perpendicular to the slope. The former is a steady force acting in the direction of the bottom of the slope, and the latter is the source of frictional force as in the previous case.

Now, let us formulate the equation of motion.

$$F=ma$$

$$x=\frac{1}{2}at^2+v_0t$$

$$v=at+v_0$$

Although sin/cos calculations are added, the calculation method remains the same as before. This time, we assume the case where the slope is converted to a slope in one dimension, so a straight uphill line and a straight downhill line can be calculated in terms of a putt.

Let’s start with the ascent. Assuming a slope angle of +2° and a stimpmeter s=9, the coefficient of friction μ=0.1. Assuming initial ball speed v0 = 2 m/s, the time t until the ball comes to a stop is

$$t=\frac{2}{0.1*9.8}=1.96秒$$

and stops after 1.96 s. From Eq. 2, it follows that

$$x=\frac{1}{2}(-\mu{}g)t^2+v_0t=-0.5*0.1*9.8*1.96^2+2*1.96=2.0m$$

The ball’s initial velocity is 2m/s on a green with a stimpmeter value s = 9. On a green with a stimpmeter value of s = 9, if the ball is launched at an initial speed of 2 m/s, it will roll 2 m.

Next is the descent. The slope angle was set to -2° and other values were left unchanged.

If the putt is made with the same strength for both uphill and downhill, the difference is xxx m under these conditions.

By the way, I used the slope angle θ° because this expression is often used in high school physics test questions. Another way to describe the slope of a slope is “gradient,” which is also often heard. I am more familiar with angles, so I use angles in “Putt Physics,” but I wonder which is more common in golf. If there are many requests for slope notation, we can accommodate them.

角度 | 勾配 |

1° | 1.8% |

2° | 3.5% |

3° | 5.2% |

4° | 7.0% |

5° | 8.7% |

By the way, if the angle exceeds 3°, the ball will not stop. Of course, the threshold depends on the green speed. So, most greens have slopes in the range of 0° to 3°, which is about 0% to 5% slope.

This completes the calculation of the motion of a ball rolling down a slope (1D).