Acceleration

**Sunyata** · May 5th 2010, 07:00 AM

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.

Could someone please help?

**skeeter** · May 5th 2010, 09:17 AM

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.

Could someone please help?

$ \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 $

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