velocity problem, calculus?

**wvlilgurl** · October 6th 2008, 01:24 PM

A motorboat is moving at 30m/s when its motor suddenly quits and it is 800 m from the shore. Ten seconds later the boat has slowed to 20 m/s. Assume that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all? Can the boat reach the shore?

**icemanfan** · October 6th 2008, 01:40 PM

If the resistance is proportional to the boat's velocity, then the deceleration of the boat is also proportional to its velocity. So you have the equation $v(t) = kv'(t)$ . You have to use this equation to determine an equation for the boat's distance traveled starting at the time the boat begins to slow down.

**icemanfan** · October 6th 2008, 02:04 PM

The equation $v(t) = kv'(t)$ is a common differential equation. The solution is $v(t) = Ce^{kt}$ . Since v = 30 at t = 0, this yields $30 = Ce^{k(0)} = Ce^0 = C$ , so the equation is $v(t) = 30e^{kt}$ . Now, since v(10) = 20, we have $30e^{10k} = 20$ ;

$e^{10k} = \frac{20}{30} = \frac{2}{3}$

$10k = \ln \left(\frac{2}{3}\right)$

$k = 0.1 \ln \left(\frac{2}{3}\right)$

**wvlilgurl** · October 6th 2008, 03:18 PM

Thanks! how do I find out how far the boat coasts?

**icemanfan** · October 6th 2008, 03:37 PM

The equation for position will be the antiderivative of the velocity function, which is $s(t) = \frac{30}{k}e^{kt} + C$ . Since s(0) = 0, $s(0) = \frac{30}{k}e^0 + C = \frac{30}{k} + C = 0$ . Hence $C = -\frac{30}{k}$ and
$s(t) = \frac{30}{k}e^{kt} - \frac{30}{k}$ . The limit of this function as t approaches infinity is $\lim_{t \to \infty} \left(\frac{30}{k}e^{kt} - \frac{30}{k}\right) = 0 - \frac{30}{k}$ , and that is the distance the boat coasts.

Thread: velocity problem, calculus?

LinkBack

Thread Tools

Display

velocity problem, calculus?

Similar Math Help Forum Discussions

Instantaneous Velocity calculus

Calculus Problem - Velocity

Velocity calculus

Velocity (Calculus)

Help!World Problem:Calculus III Projectile Motion (Velocity & Acceleration)

Search Tags