What Is the Velocity and Net Force After Two Football Players Collide?

[quicksilver123]

Homework Statement

Football player 1 with a mass of 110kg is moving at 8.0m/s when he crashed into football player 2 with a mass of 105kg, moving at 12m/s [N]. During the tackle, football player 1 hangs onto football player 2, giving them the same velocity during the 0.3 second collision.

A) What is their velocity after the collision?
B) What is the net force (affecting each player) during the collision?

Homework Equations

p = momentum
v' = velocity after collision

p=mv
Δp=mv₂-mv₁

Net Force = Δp / Δt

Conservation of momentum:
m₁v₁ + m₂v₂ = m(v')

The Attempt at a Solution

m₁ = 110kg
v₁ = -8m/s

m₂ = 105kg
v₂ = 12m/s

Let [N] be positive.

A)

m₁v₁ + m₂v₂ = m(v')
110kg(-8m/s) + 105kg(12m/s) = (110kg+105kg)v'
-880kg(m/s) + 1260kg(m/s) = 215kg v'
380kg(m/s) / 215kg = v'
v' = 1.76744186 m/s [N]

I've verified this answer with the course's book. I'm including it because the answer for part B depends on the answer from part A.


B)

Net Force = Δp / Δt
Net Force = (m₁v' - m₁v₁) / (0.3 seconds)
Net Force = [(110kg(1.76744186m/s)) - (110kg)(-8m/s))] / (0.3 seconds)
Net Force = [ 194.4186046kg(m/s) + 880kg(m/s) ] / (0.3 seconds)
Net Force = (1074.4418605kg(m/s)) / (0.3 seconds)
Net Force = 3581.395349kg(m/s²)

Net Force on Player 1 = 3581.395 Newtons
Net Force on Player 2 (from Newton's 3rd LoM) = -3581.395 Newtons


I think this should be correct, but I think I'm doing something wrong. The book's answer work uses the formula :
Net Force = (m₁v₁ - m₁v' / (0.3 seconds)
for net force, which doesn't make sense to me... seeing as how a change in momentum should be found by subtracting the final value from the initial.

By the way, the book's answer for player 1 is :
-2280 N

... Which makes sense if you use their formula (the calculations are correct)... but its the formula itself that I don't understand.

Any thoughts?

 

[Doc Al]

I think this should be correct, but I think I'm doing something wrong. The book's answer work uses the formula :
Net Force = (m₁v₁ - m₁v' / (0.3 seconds)
for net force, which doesn't make sense to me... seeing as how a change in momentum should be found by subtracting the final value from the initial.

You are correct. The book makes two errors:
(1) The formula is wrong; Change in anything is always final minus initial, as you point out.

By the way, the book's answer for player 1 is :
-2280 N

The second error the book makes, which is much worse, is:
(2) They neglect the sign of the velocities. That's the only way they can get this answer.

If all they did was use initial minus final, then there answer would be the same as yours except for a sign. No big deal, since the force acts on each player in different directions anyway.

 

[quicksilver123]

Here's what they did:

For player 1:

F_Net (Δt) = mv₂ - mv₁
F_Net (0.3) = (110)(-8) - (110)(+1.77)
F_Net (0.3) = -685.3
F_Net = -2280

The actual calculation is correct, but the formula isn't... I'm not sure what they're playing at.

 

[Doc Al]

Here's what they did:

For player 1:

F_Net (Δt) = mv₂ - mv₁
F_Net (0.3) = (110)(-8) - (110)(+1.77)

The calculation is OK up to here.

F_Net (0.3) = -685.3

But this step is wrong. (Do it yourself and see!)

F_Net = -2280

The actual calculation is correct, but the formula isn't...

The calculation is not correct.

 

[quicksilver123]

Yeah my bad, that's very wrong.
Shouldn't accept their answers at face-value, lol.

So my initial answer was correct?

 

[Doc Al]

So my initial answer was correct?

Yep!

 

[quicksilver123]

Thanks !