Question

A person holds a rifle horizontally and fires at a target. The bullet has a muzzle speed of 155 m/s, and the person hears the bullet strike the target 0.70 s after firing it. The air temperature is 77°F. What is the distance to the target

Answer 1

What is the distance to the target?

We have to set up a system of equations to solve this problem, but first we need to find the speed of sound:

The speed of sound is given by (approximately):

vsound = 331.4 * sqrt((TC + 273.15) / 273.15)

To convert from 77°F to °C, use the following formula:

TC = (TF – 32) * 5/9

Therefore, 77°F in °C is:

TC = (TF – 32) * 5/9
TC = (77 – 32) * 5/9
TC = 25

Calculate the speed of sound:

vsound = 331.4 * sqrt((TC + 273.15) / 273.15)
vsound = 331.4 * sqrt((25 + 273.15) / 273.15)
vsound = 346.23 m/s

Now we can set up the system of equations. We know that the time it takes the bullet to reach the target is given by:

tbullet = x / vbullet

And the time it takes the sound to reach the listener is given by:

tsound = x / vsound

So the total time (for the bullet to reach the target and for the sound of the bullet hitting the target to reach the shooter) is given by:

ttotal = tbullet + tsound

So:

0.70 = tbullet + tsound
0.70 = x / vbullet + x / vsound
0.70 = x / 155 + x / 346.23
346.23*155*0.70 = 346.23 x + 155x

3.756*10⁴ = 501.23 x

x = 74.94 m