Question

6. A person holds a rifle horizontally and fires at a target. The bullet has a muzzle speed of 150 m/s, and the person hears the bullet strike the target 0.80 s after firing it. The air temperature is 71°F. What is the distance to the target?

Answer 1

The relation between the speed of sound in air and temperature is

Speed of sound = 20.0457 √ K

K is temperature in kelvin ;

We know that,

F = 9/5 K - 459.67 ;

71 = 9/5 K - 459.67 ;

K = 294.817 Kelvin ;

Speed of sound = 20.0457 √ 294.817 = 344.19 m/s

Actually the total time ( 0.8 sec) is the time required by the bullet to reach the target (strike the target and make a sound) plus the time required by the sound to reach back to the person.

Speed of bullet = vb = 150 m/sec
Speed of sound = vs = 344.19 m/sec
Distance travelled by both = S = ? (it is same for both)
time taken by bullet = tb
time taken by sound = ts

By the conditions of the question:
tb + ts = 0.8 => tb = 0.8 - ts
and
tb x vb = ts x vs = S
putting values
(0.8 - ts) x 150 = ts x 344.19
150*0.8 - 150ts = 344.19ts
150*0.8 = 344.19ts + 150ts
150*0.8 = 494.19ts
ts = 150*0.8/494.19 = 0.24282 sec
tb = 0.8 - 0.24282= 0.55718 sec

S = vb x tb = 150 x 0.55718 = 83.577 m
S = vs x ts = 344.19 x 0.24282 = 83.576 m