Need help with real life issue.

[Lukasha]

Homework Statement

I'm not in school and although I took plenty of physics back in high school, I've forgotten most of what I need for this. I'm not asking for anyone to do the work for me, but it's been so long I don't even know which formula I should use to solve the problem. I can do the math, I just need help with setting up the formula.

The issue is that I was in a minor accident the other day and the other driver was completely at fault, but they are lying to the insurance companies about what happened. I know they are at fault and I know I can prove it with all of the information I have. Basically I need to disprove their claim which will basically show mine to be the only viable one.

The situation they claim is that at point A their car was directly in front of mine in the same lane and we were traveling the same speed. They are claiming that I changed lanes and accelerated to nearly catch up with them at which point they changed lanes into me. The way I know I can prove that it's impossible is there was a very short distance between point A and point B which was a stop light that was currently red. We were traveling around 40-50 MPH and at that speed with no stopping, it would only take around 14.5-18 seconds to travel from A to B. I need to prove that it would be impossible for me to accelerate fast enough to not only catch up to them at the point of impact and to be able to decelerate in time to stop at the light.

The information I have is the distance from A to B, the lengths of both cars and where on the cars the damage was so I know their relative position at impact, and if needed I can probably dig up the acceleration/deceleration characteristics of my car.

Homework Equations

Honestly I know this probably has to do with some type of acceleration formula but looking at the introductory physics formula page, I don't know which one to use.

The Attempt at a Solution

I have attempted to use my google-fu to try and figure out how to figure this out, but because of how common the words are, I haven't had much luck. I'm hoping someone will take pity on me and just set up what formulas I'd need and then I can plug in the numbers and do that math. I appreciate any and all help and I hope I haven't committed any cardinal physics sins by asking for help with this.

Jeff

 

[BruceW]

I'm glad to try to help. Just to check - were the police called to the accident? Because they might have taken pictures of the skidmarks the car made. But I guess if no-one was seriously hurt, then they might not have bothered with any of that...
So anyway, I'm going to assume the other driver's story, then try to disprove it:
both cars initially at speed $v_0$. Your car begins to accelerate. Assuming roughly constant acceleration:
$$v = v_0 + at$$
a is the acceleration of your car. (importantly: it is acceleration that happens even after taking air resistance into account).
t is the time from when you started accelerating, v is the speed of your car at time t.
Then, integrating again,
$$x = x_0 + v_0 t + \frac{1}{2} a t^2$$
Where $x_0$ is the initial position of your car. And x is position of your car at time t.
Now, assuming the other car is traveling at the same constant velocity this whole time, then
$$x_d = x_0 + v_0 t$$
(Where $x_d$ is the other driver's position at time t). So to find out if it were possible for you to overtake, you must use your position minus his position to find the relative change in position of the two cars.
$$x_{rel} = \frac{1}{2} a t^2$$
Where $x_{rel}$ is the change in relative position of the two cars.
Now, you said that there was a given distance the other car travelled, a to b. This will equal $v_0 t$ and since you know this distance, you can calculate the time you had to accelerate. Using this value for t in the equation for $x_{rel}$ will give the relative change in position of the two cars (i.e. how much you would have overtaken by).

So, you need to find a - the actual acceleration (after taking into account air resistance) of the car when it is traveling at $v_0$ (since the car's max possible acceleration will depend on its speed).
Also, keep in mind that I assumed constant acceleration for these equations. But as the car speeds up, acceleration will go down a bit. (I've assumed this change is too small to have noticeable effect, but this isn't necessarily true..)

 

[BruceW]

Actually, I've given in my equations that the two cars started side-by-side, but the problem was that your car started behind the other car.
But you can still use $x_{rel}$ to calculate the relative change in positions. So for you to have overtaken, it would mean $x_{rel}$ would need to equal at least the length of your car + the length of his car.

 

[Quinzio]

it would only take around 14.5-18 seconds to travel from A to B. I need to prove that it would be impossible for me to accelerate fast enough to not only catch up to them at the point of impact and to be able to decelerate in time to stop at the light.

I'll give my impression on this situation.
First of all we don't know how far is A to B.
Next, I've understood that your main concern is to show a car cannot accelerate that much in a given distance.

I'll take as example my car, which is a small Renault Clio, 1.2 liters.
Somewhere on the datasheet it's written the max power is 50kW.
Let's assume during an acceleration an average power of 30kW is delivered to the wheels.
Let's assume this average power is delivered for a 10 s, so that a total of 300kJ of energy is released.
Taking the classic ${1 \over 2}m v^2$, and a mass of 1000kg, a speed of 24.5 m/s or 88km/h is reached.
Distance traveled at this max acceleration is $v_{max} t /2$, giving a 122 meters in this case.
As it is used to say that an average car can go from 0 to 100km/h in 10s, I think this data can be quite good and realiable.
Of course friction is not taken in account here, and the road conditions, and... and ... and...
this is only to have a rough idea of what a car can do.

 

[Ray Vickson]

Homework Statement

I'm not in school and although I took plenty of physics back in high school, I've forgotten most of what I need for this. I'm not asking for anyone to do the work for me, but it's been so long I don't even know which formula I should use to solve the problem. I can do the math, I just need help with setting up the formula.

The issue is that I was in a minor accident the other day and the other driver was completely at fault, but they are lying to the insurance companies about what happened. I know they are at fault and I know I can prove it with all of the information I have. Basically I need to disprove their claim which will basically show mine to be the only viable one.

The situation they claim is that at point A their car was directly in front of mine in the same lane and we were traveling the same speed. They are claiming that I changed lanes and accelerated to nearly catch up with them at which point they changed lanes into me. The way I know I can prove that it's impossible is there was a very short distance between point A and point B which was a stop light that was currently red. We were traveling around 40-50 MPH and at that speed with no stopping, it would only take around 14.5-18 seconds to travel from A to B. I need to prove that it would be impossible for me to accelerate fast enough to not only catch up to them at the point of impact and to be able to decelerate in time to stop at the light.

The information I have is the distance from A to B, the lengths of both cars and where on the cars the damage was so I know their relative position at impact, and if needed I can probably dig up the acceleration/deceleration characteristics of my car.

Homework Equations

Honestly I know this probably has to do with some type of acceleration formula but looking at the introductory physics formula page, I don't know which one to use.

The Attempt at a Solution

I have attempted to use my google-fu to try and figure out how to figure this out, but because of how common the words are, I haven't had much luck. I'm hoping someone will take pity on me and just set up what formulas I'd need and then I can plug in the numbers and do that math. I appreciate any and all help and I hope I haven't committed any cardinal physics sins by asking for help with this.

Jeff

Lots of details are needed. To see some of the issues involved, look at the Transport Canada web page that deals with similar issues (in connection with the design of level-crossing approaches). It gives some formulas, graphs, etc., for a list of vehicles of different size, weight, configuration, etc., as well as table of the variation in the coefficient of friction with speed. Of course, they are describing a somewhat different problem, but you may get the idea that the problem is perhaps more complex than a homework problem in Physics 101. See
http://www.tc.gc.ca/eng/railsafety/guideline-rtd10-325.htm

RGV