Calculating the Fastest Car Speed with Distance and Time | Homework Help

[TFM]

Homework Statement

A car enthusiast claims to have the “fastest Fiesta in Farnham”. From a standing start, the car covers a quarter of a mile in 10.5 seconds. Making reasonable assumptions, estimate how fast the car is going by the quarter-mile post. Is your calculation likely to be an underestimate or an overestimate?

Homework Equations

$$v^2 = u^2 + 2as$$

$$v = u + at$$

The Attempt at a Solution

This seems easy, but both ways I try I find myself thinking too much...

I was first going to go with using:

$$v^2 = u^2 + 2as$$

and inserting a acceleration in, but we aren't given an acceleration (although the question asks to use reasonable assumptions), but I realized that this wouldn't take into account the time of 10.5seconds.

I then thought about using:

$$v = u + at$$

But get the same problem of not utilising the distance travelled.

What would be the best way to approach this problem. Could it be to rearrange one of the two equations I have noted above, into terms of a (ie a = ...), and then insert this into the other equation, thus utilising both the time and distance?

Many thanks in advanced,

TFM

 

[dontdisturbmycircles]

The reasonable assumption is that the acceleration is constant, all our lovely equations for constant acceleration are useless if the acceleration is not constant!

Since we are given distances and times, we can use one of our kinematics equations to find the acceleration. $$x=X_o+Vo(t)+1/2a*t^2$$

Can we cancel any of those terms?

 

[TFM]

$$x = X_o + Vo(t) + 1/2a*t^2$$

What is the Large X? is this the initial Speed?

 

[dontdisturbmycircles]

Yup!

 

[TFM]

Well, I can't see anything that cancels

$$x = X_o + Vo(t) + 1/2a*t^2$$

And the thing we don't know is Vo(t) and a.

 

[dontdisturbmycircles]

-From a standing start

 

[TFM]

Well, if the initial velocity = 0, x0 will dissapear...

 

[Kruum]

$$x = X_o + Vo(t) + 1/2a*t^2$$

What is the Large X? is this the initial Speed?

Just to make things sure the $$x_o$$ is the starting place of the car and $$v_o$$ is the initial velocity.

 

[dontdisturbmycircles]

No it won't! =-) Here is another thing we know

X-Xo = 1/4mile

You should only have one unknown, you know t, you know X-Xo, you know Vo=0, so find a! :)

 

[dontdisturbmycircles]

Yea, thanks Kruum, I kinda assumed that he had come across this equation before but perhaps not, it is one of the three widely used constant acceleration kinematics equations =-).

 

[TFM]

Okay, so:

$$(x - x_0) = 1/4miles$$

$$x - X_o = Vo(t) + 1/2a*t^2$$

$$1/4 = 0 + 1/2a*(10.5)^2$$

1/4 mile = 402m

$$402 = 0 + 1/2a*102$$

$$804 = a*102$$

$$a = 7.9 m/s^2$$

Does this look okay?

(the Kinematic equations I know are:

$$v = u + at$$

$$v^2 = u^2 + 2as$$

$$s = ut + \frac{1}{2}at$$

$$s = \frac{v +u}{2}t$$)

 

[dontdisturbmycircles]

Looks great! So now you know acceleration, now find the speed at the finish line.

Also, don't forget to answer the other question. What do you think about our assumption that the acceleration was constant? Do you think the acceleration increased with time or decreased with time? How would this affect our answer?

 

[dontdisturbmycircles]

Except I get a =7.29m/s^2, probably just a typo on your part though :D.

 

[TFM]

Yeah, that was a small rounding error. I get 7.29m

So now, put values into:

$$v = u + at$$

$$v = 7.29*10.5$$

v = 76.6 m/s = 171m/hr

Very Fast.

Also, for the second part, I say that the acceleration will decrease throughout, not constant, which would mean that this value is a overestimate.

 

[dontdisturbmycircles]

Good! =-)

 

[TFM]

Excellent.

Thanks for your assistance, dontdisturbmycircles

Most appreciated.

TFM

 

[dontdisturbmycircles]

You might want to give a reason for the acceleration decreasing, it is possible that it increases - although I tend to agree with you! Also make sure you converted m/s to km/hr properly, I get 275.76 km/h.

 

[dontdisturbmycircles]

No problem! =-)

 

[TFM]

No, that is 171 miles/hour

As for a reason for the decceleration,

As the car speeds up, the air resistance/friction acting upon it will increase, meaning that the engine will need to do more work to accelrate the car.

Does this sound okay?

 

[dontdisturbmycircles]

Yea, that sounds great =-).

 

[TFM]

Excellent

Thanks.