How Long Until Two Accelerating Cars Are 200m Apart?

[1irishman]

1. Homework Statement
Two cars are parked beside each other, facing in opposite directions. Car 1 accelerates at 5m/s^2 up to a max. speed of 25m/s, and Car 2 accelerates at 4m/s^2 up to a max. speed of 24m/s. How much time has elapsed when the cars are 200m apart?


2. Homework Equations
v=d/t
vf^2=vi^2+2ad
vf=vi+at


3. The Attempt at a Solution
Car 2
------
576=8d
d=72m
it takes car 2 6seconds to travel 72m

Car1
--------
625=10d
d=62.5m
it takes car 1 5seconds to travel 62.5m

So, the cars are 134.5m apart at this stage. They need to travel 200m-134.5m=65.5m before they are 200m apart, yes? The answer is 6.83 seconds, but i don't understand how they got that. please help?

 

[jegues]

Why you posted another thread for the same problem I have no idea, anyways...

You have two cars, they both have accelerations (atleast until they reach their maximum speed).

Use the formula V = Vo + at to figure out the times it takes each car to reach its max speed.

Once you have those figured out you can look at the distance that each car traveled in the given times.

Once you've established the point in time at which BOTH cars have reached their max speeds THEN you can simply solve the remaining distance left to by using d = v/t. (They've both reached there maximum speeds at this point, that is to say, a=0)

Hint for the last part,

If I say I have one car traveling 15m/s to the left and another car traveling 10m/s to the right, how would you keep the rate at which the two objects are being separated the same if ONE of the cars had a new speed of 0m/s?

 

[1irishman]

So are my times wrong above?

 

[jegues]

Yes. Initially you don't have an acceleration of 0. So you can't simply use v = d/t.

 

[1irishman]

okay so
Car 2
24=0+4t
t=6 is wrong to reach distance of 72m?

 

[jegues]

You computed that it takes car 2, 6 seconds to reach its maximum speed that part is correct.

Within those 6 seconds in which car 2 is approaching it's maximum speed it is also moving a distance, compute that distance.

Do the same for car 1.

 

[1irishman]

car 2
d=1/2(4)(6)^2
d=72m
car 1
d=1/2(5)(5)^2
d=62.5m
Right so far?

 

[jegues]

Looks good.

So, 5 seconds have passed, car 1 has reached its max speed but car 2 hasn't (it still has 1 second to go). Can you figure out how far car 1 is going to travel in that 1 second while car 2 is still accelerating?

 

[1irishman]

Okay, now this is where i am confused. How do we know that car 1 has reached its max speed and car 2 hasn't? I don't understand the 1 second to go part?

 

[jegues]

You computed 2 times. These are the times in which it takes each of the cars to reach their maximum speed. It took car 1 5 seconds and car 2 6 seconds.

So after 5 seconds have gone by car 1 is at its max speed and car 2 isn't.

How do you know this? Well you just told me it takes car 2 6 seconds to get to its maximum speed, its only been 5 seconds! We have still have 1 second to go!

 

[1irishman]

Oh I see, so then does this make sense for 1 second distance car 2?
d=1/2(4)(1)^2
d=2m

 

[jegues]

No. the first distance you calculated was correct, you need to find the distance car 1 will travel in that extra 1 second.

HINT: You're acceleration for car 1 is now 0.

 

[1irishman]

I think car 1 in 1 second travels d=vt. So 25m. Yes?

 

[jegues]

d= v/t

 

[jegues]

yes so car 1 goes 62.5 + 25m in 6 seconds and car 2 goes 72m in 6 seconds, both cars are at constant speed the rest should be easy.

 

[1irishman]

I have the two cars 159.5m apart. So 200m - 159.5m = 40.5m left until they are 200m apart.
I still don't understand how they got 6.83 seconds?

 

[jegues]

So in 6 seconds the cars are 159.5m apart that means they still have 40.5m left.

Well what do we know about the cars up to now?

Well they are both traveling at their maximum speeds (a=0) and they are going in opposite directions. Think you can use this to figure out how long the remaining 40.5m is going to take them?

If I say I have one car traveling 15m/s to the left and another car traveling 10m/s to the right, how would you keep the rate at which the two objects are being separated the same if ONE of the cars had a new speed of 0m/s?

 

[cepheid]

I have the two cars 159.5m apart. So 200m - 159.5m = 40.5m left until they are 200m apart.

Yeah, you're exactly right so far. All you need to do is figure out how much longer it takes (after 6 seconds) for the separation to increase by 40.5 m. In this time, t, car 1 will travel a distance d₁ = v_max1t and car 2 will travel a distance d₂ = v_max2t

The separation is the sum of these distances:

40.5 m = d₁ + d₂ = (v_max1 + v_max2)t

Solve for t and you will find out how much longer they have to go for. You already know the answer, so if you don't get t = 0.83 s, you know something went wrong.

 

[1irishman]

since in opp. directions then they each travel 20.25m
t=20.25/25=0.81 and t=20.25/24=0.84
so 6seconds plus 0.84 is 6.84seconds?

 

[jegues]

They don't each travel equal distances... If one car is moving faster than the other how could they possibly travel equal distances in the same amount of time.

I wanted to you recognize, using your intuition, that when you have two cars moving away from each other at a constant speed it would be the same as having a car move away from a stationary point(or car, in this case) at the rate of the 2 velocitys combined!

Forget the answer right now, sit down and think about it and see if you can understand it!

 

[cepheid]

I wanted to you recognize, using your intuition, that when you have two cars moving away from each other at a constant speed it would be the same as having a car move away from a stationary point(or car, in this case) at the rate of the 2 velocitys combined!

Yeah, now that I think about it, I shouldn't have given that away in my post. Understanding the *why* is what is most important. Sorry about that.

 

[1irishman]

I see how this works now. Thank you both for your help.

 

[1irishman]

I see how this works now. Thank you both for your help.