Avoiding vehicular rear-end collision

[PhizKid]

Homework Statement

A car A traveling at 161 km/hr is 676 m directly behind another car B traveling at 29.0 km/hr.
What must the deceleration be in order for car A to avoid collision with car B?

Homework Equations

Not sure

The Attempt at a Solution

So the initial velocity of car A is 161 km/hr and we're trying to find the deceleration (or negative acceleration as I'll make it here) of it. Assuming car B has no acceleration, the distance gap between them will be: 676 m + 29.0 km/hr - (161 km/hr + negative acceleration).

I tried to calculate it manually by finding the distance every second, but it takes too long on an exam and I can't find any formula that fits this criteria of information. How do I find the formula for the exact distance at a given point in time given this information?

 

[cmmcnamara]

Can you setup the equations of motion for each car? This a constant acceleration problem so the basic constant acceleration kinematic equations will apply.

 

[CAF123]

Set up a coordinate system with car A at the origin and car B 676 units ahead of it on the x axis. This will help you in writing up the equations for their positions as functions of time.
Can you do the rest?

 

[azizlwl]

I think the question should be the minimum deceleration since the harder the brake applied the sooner the car attain speed less than the front car.

 

[PhizKid]

Car A is going at:
161 km/hr - a*t

and car B is just going at:
29.0 km/hr

so the position of car A is just:
161 km/hr - a*t + 0 m.

and car B's position is:
29.0 km/hr + 676 m.

Right?

 

[CAF123]

Car A is going at:
161 km/hr - a*t

and car B is just going at:
29.0 km/hr

so the position of car A is just:
161 km/hr - a*t + 0 m.

and car B's position is:
29.0 km/hr + 676 m.

Right?

I think you are along the right lines.
The position of car A should be s_{xA}= 161t + \frac{1}{2}at^2
The position of car B should be s_{xB} = 29t + 676

 

[PhizKid]

I'm trying to stay away from the formulas because they mean nothing to me and their derivation by integration also doesn't make sense to me (why it happens or what the integration process means).

I can take the equations at face value, but I have the memory of a fish and can barely remember d=r*t unless I actually think about the values.

 

[azizlwl]

For minimum deceleration,

a=(v²-u²)/2s
a=(29²-161²)/2x676
a=-18.55m/s²

 

[CAF123]

For minimum deceleration,

a=(v²-u²)/2s
a=(29²-161²)/2x676
a=-18.55m/s

Are you sure this is correct? What I thought of doing was to set s_{xB}=s_{xA},however this equation contains two unknowns, a and t. I need to find another equation, which at the moment I am still trying to find.

 

[azizlwl]

Are you sure this is correct? What I thought of doing was to set s_{xB}=s_{xA},however this equation contains two unknowns, a and t. I need to find another equation, which at the moment I am still trying to find.

You can still do it that way too.
The third equation is the final speed of the behind car should be equal to the front car to avoid the collision.

 

[CAF123]

You can still do it that way too.
The third equation is the final speed of the behind car should be equal to the front car to avoid the collision.

So, v_{fA} = 161 + at and v_{fB} = 29. Setting these equal, rearranging for t and subbing into s_{xA}=s_{xB} yields a negative acceleration of -12.9m/s²?

 

[CAF123]

Nonsense! I have used the wrong units in my last post. I now get an answer of -0.993m/s². There is a very similar question in this PowerPoint( in fact almost identical):

www.physics.ucf.edu/~roldan/classes/phy2048-ch2_new.pdf

 

[azizlwl]

For minimum deceleration,

a=(v²-u²)/2s
a=(29²-161²)/2x676
a=-18.55m/s²

I've done a mistake for above post.

676=1/2(132)t
t=10.24s
a=-132/10.24= -12.9m/s²

As in your post #11

 

[CAF123]

I've done a mistake for above post.

676=1/2(132)t
t=10.24s
a=-132/10.24= -12.9m/s²

As in your post #11

I think you made the same mistake as me first time. The 676 is in metres, while the 132 is in km/hr.

 

[azizlwl]

Thanks for pointing that out.

 

[PhizKid]

I don't understand anything you guys did, but using the given 1D Kinematic equations, I got:

x - x_0 = 1/2(v_0 + v)t

.676 - 0 = 1/2(29 + v)t
v = (1.352 - 29t) / t

for car B

and for car A:

v^2 = (v_0)^2 + 2a(x - x_0)

v^2 = (161)^2 + 2a(0 - .676)
v = sqrt(25921 - 1.352a)

So I set those equal to each other, but I still have two unknown variables (a and t), so how do I get those values?

 

[gneill]

Homework Statement

A car A traveling at 161 km/hr is 676 m directly behind another car B traveling at 29.0 km/hr.
What must the deceleration be in order for car A to avoid collision with car B?

There seems to be a certain amount of unnecessary agony taking place over this question

Quite often when there are two or more moving objects to keep track of in a problem it can be handy to perform what is known as a change of reference frame. So far you've been looking at the problem from the point of view of a theoretical observer standing still on the road. That is to say, the coordinate system being used is anchored to the road.

But what would happen if, instead, you were to look at the problem from the point of view of an observer traveling along with car B? Suppose he's looking out the rear window of car B and sees car A approaching. What initial distance and speed of approach (also called "closing speed") will he observe?

 

[PhizKid]

I'm not sure what you mean gneill.

I tried a different approach by solving for 't' first:

Car B's position will be 26t + .676

So the final position of car A will be at car B's position, therefore:

26t + .676 = 1/2(161 + 26)t

Since the initial velocity of car A is 161, and the final velocity should be the same as car B which is 26.

Solving for t reveals t = (.676 / 67.5)

That's the first part, now to find the acceleration:

The position of car A and car B must be the same:

26t + .676 = 161t - 1/2at^2

Plug in t, and a = 13,480.0295858. Obviously this is wrong as the number seems way too large, and it is a positive acceleration which means it's speeding up. Why does the solution come to be something that doesn't make any sense?

 

[gneill]

Your solution is fine... but consider what units would be associated with it. You did your calculations assuming units of km and hours. So what units will this acceleration have?

It might be convenient to consider converting the starting values to meters and seconds at the outset, so these sort of "surprises" don't take you unawares

Regarding my previous suggestion, an observer looking out the back of car B will see car A initially at a distance of 676 meters and approaching at speed

161 - 26 = 135 km/hr

As far as he's concerned, from his point of view car A decelerates from that speed to zero in that distance. One simple, well known equation of motion can be applied to find that acceleration.

 

[PhizKid]

13,480.0295858 km/hr^2 = 1.04012574 m/s^2

:(

 

[gneill]

Should the acceleration be positive or negative?

 

[PhizKid]

It should be negative, but I am trying to obtain a correct value first before I worry about signs. -1.04m/s^2 looks close to the answer, but it'll probably be marked wrong if it's not the exact solution

 

[gneill]

What value is your target?

Edit: By the way, you seem to have started using 26 km/hr for car B rather than the 29 km/hr specified in the original problem statement. This could be the source of your current problem.

 

[PhizKid]

The solution says -0.994 m/s^2

 

[gneill]

The solution says -0.994 m/s^2

Check the edit I added to my previous post.

 

[PhizKid]

I converted everything to meters beforehand, and the solution came out to be -1.42 m/s^2 this time...

 

[gneill]

You'll have to show all of your steps and the intermediate numbers. We can't tell what went wrong if there's nothing to see

 

[PhizKid]

Sorry, I try to write neatly on loose leaf so I can take pictures of it to show my work, but it always comes out to chicken scratch. And it takes me forever to format the LaTeX.

So here is my work:

X_B - 676 = 8.0556t + \frac{1}{2}(0)t^2 \\ X_B = 8.0556t + 676

X_A = 8.0556t + 676 = \frac{1}{2}(44.72 + 8.0556)t \\ 676 = 18.3322t \\ \textrm{Therefore } t = 36.875

8.0556 + 676 = 44.72t - \frac{1}{2}a(36.875)^2 \\ -964.9944 = 679.8828a \\ a = -1.42 m/s^2

(44.72 is the initial velocity of car A in m/s, and 8.0556 m/s is car B's constant velocity)

 

[azizlwl]

If you like to know my method.
The accelerated car displacement contains 2 parts, the triangle and rectangle with speed 161km/hr and 29km/hr
The constant part contains 676m and rectangle which is equal to the accelerated car for minimum deceleration. Actually the answer is the minimum deceleration.

So i only need to calculate the non common area.


0.676=1/2(132)t
t=0.01024hr
a=-132/0.01024= -12890.625km/hr.hr=-0.9944m/s²

add: For my post#13 there's an error(corrected here) in units multiplication as noted by CAF123.

 

[PhizKid]

Isn't that just the average acceleration?

 

[azizlwl]

Yes it an average deceleration or constant acceleration as in equation s=s0+v0t+0.5t^2.

 

[gneill]

Sorry, I try to write neatly on loose leaf so I can take pictures of it to show my work, but it always comes out to chicken scratch. And it takes me forever to format the LaTeX.

So here is my work:

X_B - 676 = 8.0556t + \frac{1}{2}(0)t^2 \\ X_B = 8.0556t + 676

X_A = 8.0556t + 676 = \frac{1}{2}(44.72 + 8.0556)t

Can you explain the above equation? You've got the the distance traveled for car A as the average of the two car velocities multiplied by time. How does that come about?

 

[azizlwl]

Sketching a graph is really helpful
As you see from the graph, the aim is to reduce the speed to less than the front car to avoid collision. It can be done just behind the front car or anywhere behind the front car

http://imageshack.us/a/img688/5505/60741537.jpg

 

[PhizKid]

Can you explain the above equation? You've got the the distance traveled for car A as the average of the two car velocities multiplied by time. How does that come about?

I used the 1D Kinematic formula x - x_0 = 1/2(v_0 + v)t. The intial velocity of the car A is 44.72 m/s, and the final velocity of car A 8.0556 m/s if it is to avoid collision with car B going at the same constant velocity of 8.0556 m/s.

 

[gneill]

I used the 1D Kinematic formula x - x_0 = 1/2(v_0 + v)t. The intial velocity of the car A is 44.72 m/s, and the final velocity of car A 8.0556 m/s if it is to avoid collision with car B going at the same constant velocity of 8.0556 m/s.

Okay, that's fine. What value did you find for t?

Edit: Never mind, I see where you calculated it above.

The value is fine. I think you just have a mathematical error in your calculations for a using that value of t back in post 28. Redo the calculation.

 

[PhizKid]

So yea, I thought I covered all my grounds yet arrived at a solution that's pretty off...how is my calculation different from the ones mentioned earlier in this thread that got pretty much the exact answer they were supposed to get?

 

[gneill]

So yea, I thought I covered all my grounds yet arrived at a solution that's pretty off...how is my calculation different from the ones mentioned earlier in this thread that got pretty much the exact answer they were supposed to get?

See my most recent edit of my last post

 

[PhizKid]

See my most recent edit of my last post

8.0556 + 676 = 44.72(36.875) - \frac{1}{2}a(36.875)^2

36.875^2 = 1359.765625
1359.765625 / 2 = 679.8828125
44.72(36.875) = 1649.05
8.0556 + 676 = 684.0556

So 684.0556 = 1649.05 - 679.8828125a

684.0556 - 1649.05 = -964.9944

So -964.9944 = -679.8828125a

-964.9944 / -679.8828125 = 1.41935401551

Damnit, so I got positive 1.41935401551 m/s as acceleration, which would mean it's entirely incorrect then.

Where did I perform the wrong calculation?

 

[gneill]

On the left hand side of the first equation, you're missing multiplying the speed of car B by time. 8.0556*t + 676 = ...

 

[PhizKid]

Oh wait I was using the wrong equation. I'm supposed to be using x - x_0 = (v_0)t + (1/2)at^2, not x - x_0 = vt - (1/2)at^2, since 44.72 m/s is the initial velocity of car A. So it does come out to -1.41935401551 m/s^2.

So the displacement of car A should equal the displacement of car B, which means I did forget to multiply the velocity of car B by the time...thanks

Edit: No, I had to use the subtraction formula with the final velocity and change the final velocity to 8.0556 m/s in the equation. This yields the correct -0.994 m/s^2 solution

Thanks