# Math Help - Acceleration

1. ## Acceleration

A plane lands on a runway at 100m/s. It then brakes with a constant deceleration until it stops 2 km down the runway. Acceleration = -k, for some positive constant k. By integrating with respect to x, find k, and find v^2 as a function of x.

2. Originally Posted by Sunyata
A plane lands on a runway at 100m/s. It then brakes with a constant deceleration until it stops 2 km down the runway. Acceleration = -k, for some positive constant k. By integrating with respect to x, find k, and find v^2 as a function of x.

$
\frac{dv}{dt} = -k
$

$
v = \frac{dx}{dt}
$

$
\frac{dv}{dt} \cdot \frac{dt}{dx} = -\frac{k}{v}
$

$\frac{dv}{dx} = -\frac{k}{v}$

$
v \, dv = -k \, dx
$

$\frac{v^2}{2} = -kx + C$

$
v^2 = -k_2 x + C_2
$

at $x = 0$ , $v = 100$ m/s

$
100^2 = C_2
$

$v^2 = -k_2 x + 100^2$

at $x = 2000$ m , $v = 0$

$0 = -k_2(2000) + 100^2$

$
k_2 = 5
$

$
v^2 = -5x + 100^2
$