A pistol that fires a signal flare gives the flare an initial speed (muzzle speed) of 185 m/s.
(a) If the flare is fired at an angle of 45° above the horizontal on the level salt flats of Utah, what is its horizontal range? You can ignore air resistance.
(km)
(b) If the flare is fired at the same angle over the flat Sea of Tranquility on the moon, where g = 1.6 m/s2, what is its horizontal range?
(km)
To find the horizontal range of the flare in both scenarios, we need to use the projectile motion equations. Let's break down the problem step by step.
(a) Salt flats of Utah:
Given:
Initial speed (muzzle speed), v₀ = 185 m/s
Launch angle, θ = 45°
Acceleration due to gravity, g = 9.8 m/s²
Step 1: Calculate the time of flight.
The time of flight will be the same for the projectile's upward and downward motion. We can use the following equation to find the time it takes for the flare to reach its highest point:
t = (v₀ * sin(θ)) / g
Substituting the given values:
t = (185 * sin(45°)) / 9.8
t ≈ 3.34 s
Step 2: Calculate the horizontal range.
The horizontal range is the distance the projectile travels in the x-direction during the time of flight. We can use the following equation:
R = v₀ * cos(θ) * t
Substituting the given values:
R = 185 * cos(45°) * 3.34
R ≈ 559 m
Since the answer is required in kilometers, we convert the distance from meters to kilometers:
R ≈ 0.559 km
Therefore, the horizontal range of the flare on the salt flats of Utah is approximately 0.559 km.
(b) Sea of Tranquility on the moon:
Given:
Initial speed (muzzle speed), v₀ = 185 m/s
Launch angle, θ = 45°
Acceleration due to gravity on the moon, g = 1.6 m/s²
Step 1: Calculate the time of flight.
Using the same equation as in step 1 of scenario (a), we find:
t = (v₀ * sin(θ)) / g
t = (185 * sin(45°)) / 1.6
t ≈ 23.05 s
Step 2: Calculate the horizontal range.
Using the same equation as in step 2 of scenario (a), we find:
R = v₀ * cos(θ) * t
R = 185 * cos(45°) * 23.05
R ≈ 3422 m
Converting the distance from meters to kilometers:
R ≈ 3.422 km
Therefore, the horizontal range of the flare over the Sea of Tranquility on the moon is approximately 3.422 km.