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Appendix
Mathematical Derivations and Excel set up…
First, we can split the velocity of the snowboarder as they take off, which I will name Vo , into its vertical and horizontal components, which I will name Vv and Vh respectively. We can then represent this in a diagram, as shown below.
Vo
Vv
Vh
Then, we can use trigonometry to state that:
𝑉 𝑉 𝑉ℎ 𝑉
=
=> 𝑉 = 𝑉 × sin
=
=> 𝑉ℎ = 𝑉 × cos
We can also split the distance travelled along the half-pipe, which I will call s , into its vertical and horizontal components, S v and S h .
Sh
Sv
So
𝑆 𝑆 𝑆ℎ 𝑆
=
=> 𝑆 = 𝑆 × sin
=
=> 𝑆ℎ = 𝑆 ×
From SUVAT: = + 1 2
2 , and assuming that there is no air resistance or friction between the
snowboard and the snow, we can gain the equations: 𝑆ℎ = 𝑉ℎ × + 1 2 × 0 × 2
=> 𝑆 × = (𝑉 × cos )
①
−𝑆 = 𝑉 × + 1 2
=> − 𝑆 × = (𝑉 × sin ) − 1 2
× − × 2
× × 2
②
The gravitational field strength and the height are both negative in the equation directly above since they are both vectors, acceleration and displacement, are both vector quantities, and are going down, therefore are negative. We can then combine these equations, as shown in the processes below, to find an equation for So which combines , , 𝑉 and g.
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