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.

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

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

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

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

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

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

$\displaystyle v^2 = -k_2 x + C_2$

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

$\displaystyle 100^2 = C_2$

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

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

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

$\displaystyle k_2 = 5$

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

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