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Chapter 3: Problem 86

A salesman is standing on the Golden Gate Bridge in a traffic jam. He is at a height of \(71.8 \mathrm{~m}\) above the water below. He receives a call on his cell phone that makes him so mad that he throws his phone horizontally off the bridge with a speed of \(23.7 \mathrm{~m} / \mathrm{s}\) a) How far does the cell phone travel horizontally before hitting the water? b) What is the speed with which the phone hits the water?

Short Answer

Expert verified
Answer: The total impact speed of the cell phone when it hits the water is approximately 44.52 m/s.

Step by step solution

01

Analyze horizontal motion

The horizontal motion of the phone is constant because there is no external force acting on it in the horizontal direction. Therefore, we can use the following equation to determine the horizontal distance traveled by the phone: Horizontal Distance = Horizontal Speed × Time In this case, the horizontal speed is given as \(23.7 \mathrm{~m/s}\).
02

Analyze vertical motion

The vertical motion of the phone is influenced by the force of gravity. As the cell phone falls freely, we can use the following equation to determine the time it takes to fall from the bridge to the water: Height = 0.5 × g × Time² Here, g is the acceleration due to gravity, which is approximately \(9.81 \mathrm{~m/s^2}\). The phone is thrown from a height of \(71.8 \mathrm{~m}\).
03

Calculate the time to fall

Solving for time in the vertical motion equation, we get: Time = \(\sqrt{\frac{2 × Height}{g}}\) Plug in the values for Height and g: Time = \(\sqrt{\frac{2 × 71.8}{9.81}}\) Time ≈ 3.82 s
04

Calculate the horizontal distance

Now that we have the time it takes for the phone to fall, we can determine the horizontal distance traveled using the horizontal motion equation: Horizontal Distance = Horizontal Speed × Time Horizontal Distance = \(23.7 \mathrm{~m/s}\) × 3.82 s Horizontal Distance ≈ 90.53 m So, the cell phone travels approximately 90.53 meters horizontally before hitting the water
05

Find the vertical speed at impact

To find the vertical speed at the time of impact, we can use the following equation: Vertical Speed = g × Time Vertical Speed = \(9.81 \mathrm{~m/s^2}\) × 3.82 s Vertical Speed ≈ 37.41 m/s
06

Calculate the total impact speed

The total impact speed of the phone will be a combination of its horizontal and vertical speeds. The horizontal speed is constant and is given as \(23.7\mathrm{~m/s}\). We found the vertical speed to be approximately 37.41 m/s. We can use the Pythagorean theorem to find the total speed at impact: Total Speed = \(\sqrt{Horizontal Speed^2 + Vertical Speed^2}\) Total Speed = \(\sqrt{(23.7)^2 + (37.41)^2}\) Total Speed ≈ 44.52 m/s Thus, the phone hits the water with an approximate speed of 44.52 m/s.

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