Kinematics- Exercise: Driving a car on a Motorway (1eq, 2variables)

[bolzano95]

Homework Statement

The car in front of us is driving with velocity $v_0 = 80km/h$. How big should the acceleration be to overtake the car and not cause an accident? In the moment when we start to accelerate is the car in front of us x=160m away. The length of the cars is l=4m. Neglect the time when changing the lanes.

Relevant Equations

The relevant equations are the basic kinematic equations (look at my attempt of solution).

The car in front of us (2.car): $s_2=v_0t$
The car that is accelerating (1.car): $s_1=x+2l+s_2= \frac{1}{2} at^2$

Now, if we substitute the equations, we get $x+ 2l+ v_0t= \frac{1}{2} at^2$.
I have now 1equation with 2variables (a, t)- any suggestions on how to continue?

 

[jbriggs444]

It appears that you have assumed that the chase car begins at rest and that the 160 meter separation is measured from the rear bumper of the front car to the front bumper of the chase car. It also appears that you define the moment at which the overtake is complete as the point when the rear bumper of the chase car passes the front bumper of the front car. Finally, it appears that you have assumed constant acceleration.

Right so far?

Personally, I would have taken the comment about "changing lanes" as indicating that the cars start with matching speeds. And I would have taken the initial condition of "160 m away" as being center to center rather than rear bumper to front bumper.

You are correct that any constant forward acceleration at all will succeed in eventually overtaking the car in front.

Possibly "without causing an accident" means something. Is there a safe maximum speed limit? Or a maximum allowed time to overtake? Any context that you have not revealed?

 

[bolzano95]

jbriggs444,

Everything stands correct in your reply.

"Without causing an accident" I interpret as the rear bumper of the chase car does not bump into the front bumper of the front car.
The problem does not specify anything about safe maximum speed limit or maximum allowed time to overtake, therefore as you indicated I can choose 1variable as a free parameter and solve the equation.

This helped a lot!