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Chapter 3: Problem 37

Droplets in an ink-jet printer are ejected horizontally at \(12 \mathrm{m} / \mathrm{s}\) and travel a horizontal distance of \(1.0 \mathrm{mm}\) to the paper. How far do they fall in this interval?

Short Answer

Expert verified
To find the distance the droplet falls, calculate the time it takes for the droplet to travel the horizontal distance, then use this time interval to calculate the vertical distance fallen under gravity.

Step by step solution

01

Determine the time of flight

First, we can determine the time \( t \) it takes for the droplet to cover the horizontal distance \( x \). For a constant horizontal velocity \( v_x \), the time \( t \) can be calculated using the equation \( t = x / v_x \). Substitute \( x = 0.001 m \) and \( v_x = 12 m/s \) to find \( t \).
02

Calculate the vertical distance

Now, we can calculate how far the droplet falls vertically during the time \( t \). We can use the equation for motion under constant acceleration \( y = 0.5 * g * t^2 \), where \( g \) is the acceleration due to gravity and is approximately \( 9.8 m/s^2 \), and \( t \) is the time of flight calculated in Step 1. The initial vertical velocity \( v_y \) is zero because the droplet is fired horizontally.
03

Compute and verify the results

Next is to execute the computation for \( y \) obtained from the formula used in Step 2. Ensure that the result makes sense in the context of the problem. Droplets should fall a small distance given that the horizontal distance they travel is small.

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