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AI Analysis:
Given the Trump assignation scenario, we need to take into account both the supersonic speed of the bullet and the speed of sound for the shot heard after the bullet passes the microphone at the target point - the bullet passing the microphone and the report from the gun. Here’s the step-by-step process:
Define the variables:
v_b: Muzzle velocity of the bullet = 2,650 feet per second v_s: Speed of sound = 1,125 feet per second t_e: Echo time = 0.22 seconds d: Distance to the target
Set up the equations:
Time taken by the bullet to reach the target: t_b = d / v_b Time taken by the sound to reach the target: t_s = d / v_s
Relate the echo time to these times: The microphone hears the bullet passing first, and then it hears the sound of the gunshot 0.22 seconds later. Therefore, t_s - t_b = 0.22 seconds.
Set up the equation: d / v_s - d / v_b = 0.22
Solve for d: d (1 / v_s - 1 / v_b) = 0.22 d (1 / 1125 - 1 / 2650) = 0.22
Calculate the values inside the parentheses: 1 / 1125 ≈ 0.0008889 1 / 2650 ≈ 0.0003774 1 / 1125 - 1 / 2650 ≈ 0.0008889 - 0.0003774 = 0.0005115
Solve for d: d * 0.0005115 = 0.22 d = 0.22 / 0.0005115 ≈ 430.11 feet
Convert the distance to yards: d = 430.11 / 3 ≈ 143.37 yards
Therefore, the bullet traveled approximately 143.37 yards before the echo was heard.
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