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Running head: CASE ANALYSIS: THE HELMET PROBLEM
Case Analysis: The Helmet Problem
Name
Course
Date
1
CASE ANALYSIS: THE HELMET PROBLEM
2
Case Analysis: The Helmet Problem
Part 1
According to the information provided, the football player with helmet A has a velocity
of 6 ft/s and collides with the helmet of a stationary player (B). Considering how the collision is
inelastic, the following conditions must be met:
•
The total mechanical momentum of the two helmets must be the same than before the
collision
•
The total kinetic energy of the two helmets IS not the same after the collision
It is possible to write the law of conservation of the mechanical momentum as:
⃗ 𝐴,𝑜 + 𝑤𝐵 ∗ 𝑉
⃗ 𝐵,𝑜 = 𝑤𝐴 ∗ 𝑉
⃗ 𝐴,𝑓 + 𝑤𝐵 ∗ 𝑉
⃗ 𝐵,𝑓
𝑤𝐴 ∗ 𝑉
where wA and wB represent the mass of helmets A and B, VA,o and VB,o the initial speed of
helmets A and B, and VA,f and VB,f the final speed of helmets A and B.
According to the provided information, player B is stationary before the collision, which implies
that VB,o = 0 such that the previous equation is simplified into:
⃗ 𝐴,𝑜 = 𝑤𝐴 ∗ 𝑉
⃗ 𝐴,𝑓 + 𝑤𝐵 ∗ 𝑉
⃗ 𝐵,𝑓
𝑤𝐴 ∗ 𝑉
Moreover, since the two helmets have the same mass, it is possible to write this equation as:
⃗ 𝐴,𝑜 = 𝑤 ∗ (𝑉
⃗ 𝐴,𝑓 + 𝑉
⃗ 𝐵,𝑓 )
𝑤∗𝑉
which is equivalent to:
⃗ 𝐴,𝑜 = 𝑉
⃗ 𝐴,𝑓 + 𝑉
⃗ 𝐵,𝑓
𝑉
It is thus possible to write,
𝑉𝐴,𝑜𝑥 = 𝑉𝐴,𝑓𝑥 + 𝑉𝐵,𝑓𝑥
and
𝑉𝐴,𝑜𝑦 = 𝑉𝐴,𝑓𝑦 + 𝑉𝐵,𝑓𝑦
CASE ANALYSIS: THE HELMET PROBLEM
3
On the other hand, the coefficient of restitution, e, is:
𝑒=
𝑉𝐵,𝑓𝑥 − 𝑉𝐴,𝑓𝑥
= 0.6
𝑉𝐴,𝑜𝑥
Substituting from the previous equation,
𝑒=
𝑉𝐵,𝑓𝑥 − 𝑉𝐴,𝑓𝑥
= 0.6
𝑉𝐴,𝑓𝑥 + 𝑉𝐵,𝑓𝑥
Operating,
𝑉𝐵,𝑓𝑥 − 𝑉𝐴,𝑓𝑥 = 0.6 ∗ (𝑉𝐴,𝑓𝑥 + 𝑉𝐵,𝑓𝑥 )
𝑉𝐵,𝑓𝑥 − 𝑉𝐴,𝑓𝑥 = 0.6𝑉𝐴,𝑓𝑥 + 0.6𝑉𝐵,𝑓𝑥
0.4𝑉𝐵,𝑓𝑥 = 1.6 ∗ 𝑉𝐴,𝑓𝑥
𝑉𝐵,𝑓𝑥 = 4𝑉𝐴,𝑓𝑥
Besides, we know that the collision is oblique, since...