How many times did you go sledding with your kids and watched in horror as they kept sledding down the hill and further, towards a frozen stream or perhaps your neighbor's fence? With this sled ride calculator, you will be able to assess how far – and how fast – your kid is going to ride. All you have to know is how high the hill is, and what kind of sled is your child using.
Before the ride
Before your child begins to sled down the hill, it has to climb up to the highest point. We generally assume that it has no initial speed before it starts moving down the hill. The information you have to input in the sled ride calculator is:
Hill inclination — There's a substantial difference between going down a gentle slope and a steep hill. Input the inclination of the hill, measured as the angle that the slope makes with the horizontal.
Hill height — The height of the hill at the highest point, measured vertically.
Length of slope — The total length of slope along which your child will move. If you know this value, you can input it directly and neglect the previous two.
Type of sled — Various materials behave differently when sliding on snow. You probably noticed that while wooden or metal sleds seem to go far, plastic ones don't work too well on snow. If your sled is made of more than one material, check the surfaces that are in contact with snow. The type of sled influences the coefficient of friction. Our calculator uses the following values (source):
- Waxed wood on wet snow: μ = 0.10
- Waxed wood on dry snow: μ = 0.04
- Plastic on snow: μ = 0.30
- Metal on snow: μ = 0.03
Phase 1: sledding down the slope
When the child begins sliding down the slope, its motion can be compared to a rigid body sliding down a plane. There are three main forces acting on it (neglecting air resistance):
- Gravity — This force acts downwards and is equal to the mass multiplied by gravitational acceleration (g = 9.80665 m/s²) and denoted by
- Normal force — This is the force exerted on the sled by the ground. It is perpendicular to the slope and denoted by
- Friction — Friction acts in the direction opposite to the direction of movement. It is equal to the normal force multiplied by a coefficient of friction μ, and is denoted by
The resultant force will cause acceleration down the slope. Surprisingly, this acceleration is not dependent on the mass of the child.
You can try to calculate the resultant force yourself. If you divide it by the mass, you will get acceleration that can be found according to the formula:
acceleration = g * [sin(Θ) - cos(Θ) * μ]
Once you know the acceleration, you can find the time it takes to reach the bottom of the hill, as well as the maximum speed reached during sledding:
time = √[2 * slope length / acceleration]
final speed = acceleration * time
Phase 2: traveling after the slope
After the sled reaches the bottom of the hill, it begins to decelerate until it comes to a complete stop. That is because the only force that now acts on the sled is the friction, directed opposite to the direction of motion. Knowing the speed that the sled gained during the descent down the hill, you can easily calculate the deceleration and the time it takes for the sled to come to a complete halt.
deceleration = - g * μ
time to stop = final speed / deceleration
You can also calculate the distance that the sled will travel from the bottom of the hill before stopping, using the following equation:
distance = - final speed²/ (2 * deceleration)
Remember that the calculated results don't take air resistance into consideration. Hence, the distance traveled by a child on a sled will always be a bit smaller in reality.