Two football players separated by 42m run directly toward each other. Football player 1 starts from rest and accelerates at 2.4m/s^2
, and football player 2 moves uniformly at 5.4m/s. How long does it take for the players to collide?
Given: player 1: v1=0m/s, a=2.4m/s^2
Player 2: v1=v2=5.4m/s, a=0m/s
Conventions: right= positive, left= negative.
Note: I did not plug in 42m for Δd because neither of the players are actually displaced 42m as they end up colliding.
I used the SUVAT/5 kinematics equations, one equation for each player.
For player 1: Δd=v1Δt + 1/2aΔt^2
After plugging in the given values: Δd=1/2(2.4)Δt^2.
After simplifying: Δd=1.2Δt^2
For player 2: Δd= [(v1+v2)/2]Δt
After plugging in the given values: Δd=[(-5.4-5.4)/2]Δt
After simplifying: Δd=-5.4Δt
The Attempt at a Solution
So, I took my 2 simplified equations and set them equal to each other. My goal was to solve for Δt. I thought this would work because in math class, we found the point of intersection of two linear functions by isolating for y in both and then setting them equal to each other. We then solved for x and plugged that value back into one of the equations and solved for y.
Both of my equations were isolated for Δd so I thought it would make sense to set them equal and solve for Δt:
I used the quadratic formula to solve for Δt:
After plugging in values and simplifying (a=1.2, b=5.4):
Both of my answers are inadmissable. Why did this happen?
My teacher showed us the solution to this problem and she did the same thing I did except she used a different equation for player 2(Δd=v1Δt+1/2aΔt^2 -- This is the same equation used for player 1). Also, she plugged in 'x' for Δd in the equation for player 1 and 42-x for Δd in the equation for player 2. She then simplified both equations, set them equal and solved for Δt using quadratic formula.
Why is it wrong to not plug anything in for Δd and just set the 2 equations equal to each other, since they're both isolated for Δd? Aren't we just solving for Δt at the point where Δd or player 1=Δd of player 2 (this is the point where they collide).
In short, can someone please explain why what I did was wrong and what my teacher did was right?
Edit: Please help! My exam is in less than 10 days and it's a grade 12 class so it's very important!!!