Aiming a rifle missing the target
A rifle is aimed horizontally at a point 50 m away. The bullet hits the target 1.0 cm below the aiming point. (1) How long did the bullet travel in the air? (2) What was its muzzle velocity? We neglect the influence of a hindering force due to the air.
Data and unknown:
s = 50.0m - distance traveled by the bullet in horizontal direction
ΔH = 1.0cm displacement in vertical direction
t = ? time of flying from rifle to target
v0 = ? initial velocity
g = 9.81m/s2 as usual we assume knowledge of gravitational acceleration, unless the problem requires the calculation of this acceleration from given data.
Solution.
The motions of a bullet in horizontal and vertical directions are independent. During the 50 m flight the bullet falls 1.0cm downwards due to gravitational acceleration. This can be expressed by the formula
ΔH = (1/2)gt2 (1)
The horizontal displacement is governed by the formula
s = v0t (2)
because in projectile motion horizontal velocity is constant. That is why we assumed no air resistance.
Equations (1) and (2) can be solved with respect to time t and velocity v0,
From formula (1)
t = (2ΔH/g)1/2 (3)
Remember, that notation to the power 1/2 means square root of..
From formula (2) we have
v0 = s/t
and after substituting t from (3), we get
v0 = s/ (2ΔH/g)1/2
Substituting numbers given in the problem we have
t = 0.0452 s , v0 = 1107 m/s