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Answer :
The lost kinetic energy in the collision between a safety and a wide receiver can be calculated using the principles of conservation of momentum and the kinetic energy formula, taking into account the conversion of weight to mass in slugs and yards to feet for velocity.
To calculate the kinetic energy lost in the collision, we first need to find the initial kinetic energy of the system, then the final kinetic energy after the collision, and finally, subtract the final kinetic energy from the initial kinetic energy.
The initial kinetic energy [tex](\( KE_i \))[/tex] of the system before the collision is the sum of the kinetic energies of the safety and the wide receiver:
[tex]\[ KE_i = \frac{1}{2} m_{\text{safety}} v_{\text{safety}}^2 + \frac{1}{2} m_{\text{receiver}} v_{\text{receiver}}^2 \][/tex]
Given:
Mass of the safety [tex](\( m_{\text{safety}} \))[/tex] = 195 lb
Mass of the wide receiver [tex](\( m_{\text{receiver}} \))[/tex] = 180 lb
Velocity of safety [tex](\( v_{\text{safety}} \))[/tex] = 12.8 yd/s
The velocity of the wide receiver [tex](\( v_{\text{receiver}} \))[/tex] = 9.1 yd/s
First, we need to convert the velocities from yards per second to feet per second to match the unit of the conversion factor provided (1 slug ft²/s² = 1.3558):
[tex]\[ 1 \text{ yd/s} = 3 \text{ ft/s} \][/tex]
So, [tex]\( v_{\text{safety}} = 12.8 \times 3 = 38.4 \) ft/s and \( v_{\text{receiver}} = 9.1 \times 3 = 27.3 \) ft/s.[/tex]
Now, we can calculate the initial kinetic energy:
[tex]\[ KE_i = \frac{1}{2} \times 195 \times (38.4)^2 + \frac{1}{2} \times 180 \times (27.3)^2 \][/tex]
[tex]\[ KE_i = \frac{1}{2} \times 195 \times 1479.36 + \frac{1}{2} \times 180 \times 745.29 \][/tex]
[tex]\[ KE_i \approx 143436.24 + 67193.2 \][/tex]
[tex]\[ KE_i \approx 210629.44 \text{ ft²/s²} \][/tex]
Next, after the collision, the two players move together as one entity. Thus, the final kinetic energy [tex](\( KE_f \))[/tex] is:
[tex]\[ KE_f = \frac{1}{2} (m_{\text{safety}} + m_{\text{receiver}}) v_{\text{final}}^2 \][/tex]
Where [tex]\( v_{\text{final}} \)[/tex] is the final velocity of the combined system.
To find [tex]\( v_{\text{final}} \[/tex], We can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision:
[tex]\[ m_{\text{safety}} \times v_{\text{safety}} + m_{\text{receiver}} \times v_{\text{receiver}} = (m_{\text{safety}} + m_{\text{receiver}}) \times v_{\text{final}} \][/tex]
Substituting the given values:
[tex]\[ (195 \times 38.4) + (180 \times 27.3) = (195 + 180) \times v_{\text{final}} \][/tex]
[tex]\[ 7488 + 4914 = 375 \times v_{\text{final}} \][/tex]
[tex]\[ 12402 = 375 \times v_{\text{final}} \][/tex]
[tex]\[ v_{\text{final}} = \frac{12402}{375} \][/tex]
[tex]\[ v_{\text{final}} \approx 33.204 \text{ ft/s} \][/tex]
Now, we can calculate the final kinetic energy:
[tex]\[ KE_f = \frac{1}{2} \times (195 + 180) \times (33.204)^2 \][/tex]
[tex]\[ KE_f = \frac{1}{2} \times 375 \times 1102.11 \][/tex]
[tex]\[ KE_f = \frac{1}{2} \times 413081.25 \][/tex]
[tex]\[ KE_f \approx 206540.625 \text{ ft²/s²} \][/tex]
The kinetic energy lost in the collision is the difference between the initial and final kinetic energies:
[tex]\[ \text{Kinetic energy lost} = KE_i - KE_f \][/tex]
[tex]\[ \text{Kinetic energy lost} = 210629.44 - 206540.625 \][/tex]
[tex]\[ \text{Kinetic energy lost} \approx 4088.815 \text{ ft²/s²} \][/tex]
Now, let's convert this to slug ft²/s² using the provided conversion factor:
[tex]\[ \text{Kinetic energy lost} \approx 4088.815 \times \frac{1}{1.3558} \][/tex]
[tex]\[ \text{Kinetic energy lost} \approx 3016.18 \text{ slug ft²/s²} \][/tex]
Finally, let's round this to one decimal place:
[tex]\[ \text{Kinetic energy lost} \approx \boxed{3016.2} \text{ slug ft²/s²} \][/tex]
So, the kinetic energy lost in the collision is approximately 3016.2 slug ft²/s².
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