Question

The force of resistance on an object moving through a fluid is proportional to the
square of its velocity. The initial velocity of an object of mass 1 kg moving through the
fluid is 40 ms−1. After 5 s the velocity reduces to 20 ms−1. Calculate its velocity after
10 s

Answer 1

Question #85562, Physics / Other

The force of resistance on an object moving through a fluid is proportional to the square of its velocity. The initial velocity of an object of mass $1\mathrm{kg}$ moving through the fluid is $40~\mathrm{ms^{-1}}$. After 5 s the velocity reduces to $20~\mathrm{ms^{-1}}$. Calculate its velocity after 10 s

Solution

$$F = m a = m \frac {d v}{d t} = - k v ^ {2}$$ $$\frac {d v}{d t} = - \frac {k}{m} v ^ {2}$$ $$\int \frac {d v}{v ^ {2}} = - \frac {k}{m} \int d t$$ $$- \frac {1}{v} = - \frac {k}{m} t + C$$

1) $t = 0$ s.

$$- \frac {1}{4 0} = - \frac {k}{m} (0) + C$$ $$C = - \frac {1}{4 0} \frac {s}{m}.$$

2) $t = 5$ s.

$$- \frac {1}{2 0} = - \frac {k}{1} (5) - \frac {1}{4 0}$$ $$k = 0. 0 0 5 \frac {k g}{m}.$$

3) $t = 10$ s.

$$- \frac {1}{v} = - 0. 0 0 5 (1 0) - \frac {1}{4 0}$$ $$v \approx 1 3. 3 \frac {m}{s}.$$

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