# At the End of a Slippery Slope: A Pilot Study of Deceleration Mats for Snow Tubing
## 4 Discussion
### Safety Considerations: Reduced Run-Out Distance Needed with Deceleration Mats  
The use of multiple, consecutive deceleration mats in each snow tubing lane is not uncommon. Using a constant COF for each surface (snow or deceleration mats) and the balance of linear momentum, the run-out length and number of deceleration mats necessary to bring snow tube riders to a stop can be estimated. Care must be taken when assessing snow tube lane design in general because there are several difficult to specify factors that may influence the run-out kinematics for a given rider, such as rider mass, snow condition, temperature, variables related to air resistance, the amount of snow covering the deceleration mat surface, etc. With this caveat, the value of using deceleration mats can be shown with an example. The effect of deceleration mats on the distance traveled in the run-out of a tubing hill is assessed below. To illustrate the effects of snow tube deceleration mats, we started with the balance of linear momentum for the center-of-mass of the snow tube and rider system on a horizontal (0\({}^{\circ}\)) run-out:  
\[ma=-\mu_{X}(x)\cdot mg-\ \frac{1}{2}\rho C_{d}Av^{2} \tag{3}\]  
where \(x\) is the position of the system along the run-out, \(v\) is the velocity, \(a\) is the acceleration of the system, \(m\) is the mass of the snow tube and rider system, \(g\) is the gravitational constant, \(\rho\) is air density, \(C_{d}\) is the geometric drag coefficient, \(A\) is the cross-sectional area normal to the direction of motion, and \(\mu_{x}(x)\) is the coefficient of friction that depends on the location of the snow tube and changes whether it is on snow, a flat mat, or a folded mat. Equation (3) can be rewritten to:  
\[\bar{x}+\ \frac{1}{2}\frac{\rho C_{d}A}{m}\dot{x}^{2}\ +\ \mu_{X}(x)g=0 \tag{4}\]  
and solved numerically. For this example, a snow tube rider enters a 50 m long, horizontal run-out at 9.5 m/s. The rider and snow tube system have a total mass (\(m\)) of 91 kg (75 kg snow tube rider with 16 kg of equipment, including clothing and a snow tube), with a total cross-sectional area (\(A\)) of 0.59 m\({}^{2}\) projected along the downhill direction of travel. Further consider the case in which there is no wind, the temperature is 0 \({}^{\circ}\)C, the air density (\(\rho\)) is 1.18 kg/m\({}^{3}\), and the \(C_{d}\) is 1. Three example configurations were considered: (1) _no mats_--no deceleration mats are placed in the run-out and \(\mu_{X}(x)\) is constant at 0.08; (2) _flat mats_--1.52 m long, flat mats are placed in the run-out and separated by 0.5 m, such that \(\mu_{x}\) = 0.11 while the snow tube is on a flat deceleration mat and \(\mu_{x}\) = 0.08 when the snow tube is on the snow between mats; (3) _folded mats_--the mats from _flat mats_ (2) configuration are folded at the downhill end (the front edges of the mats remain in the same places), such that the mats are effectively 1.02 m long and \(\mu_{x}=0.26\) while the snow tube is on a folded mat and \(\mu_{x}=0.08\) when the snow tube is on the snow between mats. In this example, the number of deceleration mats was the same in the _flat mats_ (2) and _folded mats_ (3) configurations, but there is more space (tube-on-snow distance) between mats in the _folded mats_ configuration because of the mat folding. Equation (4) was solved numerically (using MATLAB R2021a, Mathworks, Natick, MA, USA) for the three configurations; the velocity of the snow tube system as a function of distance travelled in the run-out is shown in Figure 4.  
This example shows the value of deceleration mats for a theoretical tubing hill. In the _no mats_ configuration, a typical snow tube and rider would require 49.1 m to stop (approximately 10.9 s), using 98% of the run-out distance and offering very little additional space in case of an unusually fast descent or a decrease in friction in the run-out; in this example, there is only a small margin to accommodate changes in environmental factors and rider characteristics that may result in faster run-out entry speeds.