A rifle with a muzzle velocity of Vo shoots a bullet at a target R ft away. How high above the
target must the rifle be aimed so that the bullet will hit the target?
R=v₀•t
t=R/v₀
h=g•t²/2= g•R²/2•v₀²
To find the height above the target that the rifle must be aimed, we first need to understand the motion of the bullet.
Assuming there is no air resistance, the horizontal motion of the bullet can be described by the equation:
R = Vx * t
where R is the distance to the target, Vx is the horizontal component of the velocity, and t is the time of flight.
We can find the time of flight by using the vertical motion of the bullet. The vertical motion is affected by the acceleration due to gravity (-9.8 m/s²). The vertical position can be described by the equation:
y = Viy * t + (1/2) * g * t²
where y is the vertical displacement, Viy is the initial vertical velocity, g is the acceleration due to gravity, and t is the time of flight.
Since the initial vertical velocity (Viy) is equal to the muzzle velocity (Vo) multiplied by the sine of the launch angle (θ), we can rewrite the equation as:
y = Vo * sin(θ) * t + (1/2) * g * t²
To find the time when the bullet reaches the target, we equate the horizontal distance (R) to the horizontal component of the velocity multiplied by the time t:
R = Vx * t
Since Vx is equal to Vo multiplied by the cosine of the launch angle (θ), we can rewrite the equation as:
R = Vo * cos(θ) * t
Next, we need to solve for t in terms of R and Vo. Dividing the equation for y by the equation for R, we get:
y/R = (Vo * sin(θ) * t + (1/2) * g * t²) / (Vo * cos(θ) * t)
Simplifying the equation, we have:
y/R = (tan(θ) * t + (1/2) * g * t²) / (cos(θ) * t)
y/R = (tan(θ) + (1/2) * g * t) / (cos(θ))
Notice that t appears in numerator and denominator on the right side of the equation. Hence, we can cancel out t:
y/R = (tan(θ) + (1/2) * g * t) / (cos(θ))
y/R = tan(θ) + (1/2) * g * t
Rearranging the equation for y, we find:
y = R * (tan(θ) + (1/2) * g * t)
Now we solve this equation for y. Rearranging, we have:
y - R * tan(θ) = (1/2) * g * R * t
Simplifying further:
y - R * tan(θ) = (1/2) * g * R * t
Finally, solve for y:
y = R * tan(θ) + (1/2) * g * R * t
By substituting known values for R, Vo, g, and solving for y, we can find the height above the target that the rifle must be aimed.