Very beginner physics 1D motion question

[mistymoon_38]

[SOLVED] Very beginner physics 1D motion question!

Homework Statement

Two cars move in opposite directions toward each other on a level, straight, one-lane road. When both drivers begin to brake, their front bumpers are 275 meters apart. The smaller car is initially moving at 25.0 m/s and the larger is initially moving at 30.0 m/s. The magnitude of the acceleration of the smaller car is 20.0% greater than the magnitude of the acceleration of the larger. Determine the magnitude of acceleration for each car such that when their front bumpers touch, they have just come to rest

Homework Equations

(Vf)^2=(Vi)^2+2a(Xf-Xi)

The Attempt at a Solution

TRANSLATION:
1 stands for the larger car, 2 for the smaller car...
Xi1=0 Xi2=275
Xf1=unknown Xf2=unknown Xf1=Xf2
Vi1=30 Vi2=-25
Vf1=0 Vf2=0
a1=? a2=? a2=1.2a1

EQUATE:
0=(Vi1)^2+2a1(Xf1)
0=(Vi2)^2+2.4a1(Xf2-Xi2)

To solve for a1 I isolated Xf1 in the first equation and then plugged that in where Xf2 is on the second equation, since they are equal. I ended up with:

a1=((Vi2)^2-1.2(Vi1)^2)/(2.4Xi2)
a1= ((-25)^2-1.2(30)^2)/(2.4*275)
a1=-.689 m/s^2
and therefore
a2=-.827 m/s^2
which is the magnitude if you take the absolute value...

So I am wondering if this is correct becaue I am having difficulty verifying my answers and I'm not sure if this is reasonable..

 

[alphysicist]

Hi mistymoon_38,

To check this problem you can use the accelerations you found to find the displacement that each car takes during their trip and see if the distances each travels adds up to 275 meters.

There's a subtle error in one of your expressions. You have set up the problem so that for each car the positive and negative directions are the same. However, the accelerations of the cars are in opposite directions (so one is negative and one is positive). So you cannot write $a_2 = 1.2 a_1$ because this forces both accelerations to have the same sign. If you use $a_2 = -1.2 a_1$ I believe you'll get the proper relationship between the accelerations.

 

[Moetasim]

Your solution is correct. To verify your answers, you can use the equation Vf^2=Vi^2+2a(Xf-Xi) for both cars and see if they both come to rest at the same time and distance. You can also check if the acceleration for the smaller car is indeed 20% greater than the acceleration for the larger car. Additionally, you could plot the position-time graphs for both cars and see if they intersect at the same point where they both come to rest. Keep up the good work!