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Chapter 7: Problem 80

A stationary 0.1-g fly encounters the windshield of a \(1000-\mathrm{kg}\) automobile traveling at \(100 \mathrm{km} / \mathrm{h} .\) (a) What is the change in momentum of the car due to the fly? (b) What is the change of momentum of the fly due to the car? (c) Approximately how many flies does it take to reduce the car's speed by \(1 \mathrm{km} / \mathrm{h} ?\)

Short Answer

Expert verified
Short Answer: It takes approximately 100,000 flies to reduce the car's speed by 1 km/h.

Step by step solution

01

Convert units to the SI system

First, we need to convert the given quantities into the SI system. The fly's mass is given in grams, and the car's velocity is given in km/h. Convert them to kg and m/s respectively. Fly's mass: \(0.1 \mathrm{g} = 0.0001 \mathrm{kg}\) Car's velocity: \(100 \mathrm{km/h} = \frac{100 \cdot 1000}{3600} \mathrm{m/s} = 27.8 \mathrm{m/s}\)
02

Calculate initial momenta

Calculate the initial momentum of the fly and the car. Since the fly is stationary, its initial velocity is 0. Fly's initial momentum: \(p_{\text{fly,initial}} = m_{\text{fly}} \cdot v_{\text{fly,initial}} = 0.0001 \cdot 0 = 0 \mathrm{kg \cdot m/s}\) Car's initial momentum: \(p_{\text{car,initial}} = m_\text{car} \cdot v_{\text{car,initial}} = 1000\cdot 27.8 = 27800\;\mathrm{kg \cdot m/s}\)
03

Calculate change in momentum (a and b)

After the collision, the fly and the car have the same final velocity since they stick together. So to find the change in momentum, we will use the initial and final momenta. Change in momentum of the car: \(\Delta p_\text{car} = p_{\text{car,final}} - p_{\text{car,initial}} = -m_{\text{fly}} \cdot v_{\text{car,initial}} = -0.0001 \cdot 27.8 = -0.00278 \;\mathrm{kg \cdot m/s}\) Change in momentum of the fly: \(\Delta p_\text{fly} = p_{\text{fly,final}} - p_{\text{fly,initial}} = m_{\text{fly}} \cdot v_{\text{car,initial}} = 0.00278 \;\mathrm{kg \cdot m/s}\)
04

Calculate the number of flies needed to reduce car's speed by 1 km/h (c)

First, we need to find the change in momentum that corresponds to a 1 km/h reduction in the car's speed. Change in car's speed: \(\Delta v_\text{car} = \frac{1 \cdot 1000}{3600} \mathrm{m/s} = 0.2778 \mathrm{m/s}\) Change in car's momentum: \(\Delta p_\text{car} = m_\text{car} \cdot \Delta v_\text{car} = 1000 \cdot 0.2778 = 277.8 \;\mathrm{kg \cdot m/s}\) Now, we divide the change in car's momentum by the change in momentum produced by one fly. Number of flies: \(\frac{\Delta p_\text{car}}{\Delta p_\text{fly}} = \frac{277.8}{0.00278} \approx 100000\) So, it takes approximately 100,000 flies to reduce the car's speed by 1 km/h.

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