Two vehicles braking to avoid a collision

[Arne B C]

A tractor and a Tesla drives in opposite directions. The tractor has a speed of 40.0km/h and the car has a speed of 80.0 km/h. The Tesla suddenly sees the tractor, and they both immediately start braking, both with constant accelerations of 5.00 m/s2 (opposite to their directions of motion).

a) If the initial distance between the two is 60.0 m, do they hit each other? If so, where, and with what relative speed on impact? If not, what is the distance between the two when they both stop?

vi (tractor) = 40.0km/h
xi (tractor) = 0
ax (tractor) = -5.0 m/s2
vi (car) = -80.0 km/h, since it goes in the opposite direction
xi (car) = 60.0m
ax (car) = 5.0 m/s2Tried to use xf=xi + vxi*t + 0.5*ax*t^2 and do one for the tractor and one for the car, then put them equal to each other. Tried then to take the ABC-formula, but didn't get any results. And now I'm stuck.

 

[TSny]

Hello and welcome to PF!

Tried to use xf=xi + vxi*t + 0.5*ax*t^2 and do one for the tractor and one for the car, then put them equal to each other.

In this approach, you might be making an assumption that is not valid. If there is a collision, are both vehicles necessarily in motion at the time of impact?

 

[SteamKing]

A tractor and a Tesla drives in opposite directions. The tractor has a speed of 40.0km/h and the car has a speed of 80.0 km/h. The Tesla suddenly sees the tractor, and they both immediately start braking, both with constant accelerations of 5.00 m/s2 (opposite to their directions of motion).

a) If the initial distance between the two is 60.0 m, do they hit each other? If so, where, and with what relative speed on impact? If not, what is the distance between the two when they both stop?

vi (tractor) = 40.0km/h
xi (tractor) = 0
ax (tractor) = -5.0 m/s2
vi (car) = -80.0 km/h, since it goes in the opposite direction
xi (car) = 60.0m
ax (car) = 5.0 m/s2Tried to use xf=xi + vxi*t + 0.5*ax*t^2 and do one for the tractor and one for the car, then put them equal to each other. Tried then to take the ABC-formula, but didn't get any results. And now I'm stuck.

Why don't you show us your attempt at solving this problem. You may have made a mistake in your calculations.

 

[Arne B C]

Now I think I got it.
The tractor is at full stop when the collision happens, and the car is driving with a speed of 4m/s.
I might have done it in a messy way, do you guys have a faster way to do the problem?
First i solved for t, and i got that the car stops after 4.44s and the tractor after 2.22s.
I then solved the equation xf= xi+ vit + 0.5at^2 for both and found out that there must be a collision since the relative distance traveled between them was over 60 meters.
Then I used the equation: vf^2= vi^2 +2a(xf-xi) for the car to find out the speed of the car on impact (didn't solve this for the tractor because the tractor was at a full stop).

Now my problem is with the next question:
In fact, it takes both of them 0.50 s to react to seeing each other, so they only start braking 0.50s after the car comes round the corner.

b) What is the answer to the questions in a) in this case?

So first I took their speed times 0.5s, and got that the car drove 11.1m before breaking, and the tractor drove 5.55m.
So now xi for the tractor is = 5.55m, and xi for the car is (60-11.1)m=48.9m.
Then I'm lost...

 

[TSny]

Now I think I got it.
The tractor is at full stop when the collision happens

OK, but I don't think you have shown how you know that the tractor has stopped before the collision occurs.

and the car is driving with a speed of 4m/s.

If you are going to include 3 significant figures, then 4 m/s is not precise enough.

b) ...
So first I took their speed times 0.5s, and got that the car drove 11.1m before breaking, and the tractor drove 5.55m.
So now xi for the tractor is = 5.55m, and xi for the car is (60-11.1)m=48.9m.

OK

Then I'm lost...

You should be able to repeat the same sort of analysis as used in part (a).