Physic problem about speed

Aug 2014
115
1
America
A certain airport runway of length L allows planes to accelerate
uniformly from rest to takeoff speed using the full length of
the runway. Because of newly designed planes, the takeoff
speed must be doubled, again using the full length of the runway
and having the same acceleration as before. In terms of L,
what must be the length of the new runway?

If the speed double the length of the runway has to double too right?
 
Apr 2015
263
57
Somerset, England
If the speed double the length of the runway has to double too right?
Does it?

This is a problem in kinematics.
Have you heard of this equation?

v2 = u2 + 2as
 
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Aug 2014
115
1
America
nope :( can you explain to me why the runway quadruple?
 

romsek

MHF Helper
Nov 2013
6,838
3,079
California
nope :( can you explain to me why the runway quadruple?
starting with constant acceleration $a$, initial velocity $v(0)=0$, and initial starting point $s(0)=0$ we have

$v(t)=a t$

$s(t)=\dfrac {a t^2}{2}$

Now say the plane has to accelerate to $v_{to}$ to take off. This occurs at

$v_{to} = a t_{to}$

$t_{to} = \dfrac {v_{to}}{a}$

and the plane has travelled

$s=s(t_{to})=\dfrac a 2 t_{to}^2=\dfrac {v_{to}^2}{2a}$

If we now double the takeoff speed we see that

$s = \dfrac {(2v_{to})^2}{2a}=\dfrac{4 v_{to}^2}{2a} = 4 \times \dfrac{v_{to}^2}{2a}$

The factor on the right is the length of runway needed for takeoff velocity $v_{to}$ and so we see doubling this leads to a factor of 4 increase in the runway length.