a
ufo is trying to escape by flying at the constant velocity of 70m/s at an upward angle of 2 degrees. it passes directly over you at a height of 40 m. at the instant it is directly above you you attempt to shoot it down by launching a projectile.
what is the lowest launch speed for initial velocity you could use to successfully hit the ufo? and at what angle would you launch at?
To determine the lowest launch speed and launch angle required to hit the UFO, we can use basic projectile motion equations. Here's how you can calculate it:
Step 1: Analyze the motion of the UFO.
The UFO is flying at a constant velocity of 70 m/s at an upward angle of 2 degrees. Since the velocity is constant, we can assume there is no horizontal acceleration.
Step 2: Consider the motion of the launched projectile.
The projectile you launch will follow a curved path, influenced by both gravity and the initial launch conditions.
Step 3: Calculate the time of flight.
Since the UFO passes directly over you at a height of 40m, we can determine the time it takes for the projectile to reach the same height. This will be our time of flight (t).
Using the vertical motion equation:
Δy = V₀y * t + 0.5 * a * t²,
where Δy is the change in height (40m), V₀y is the initial vertical velocity of the projectile, a is the acceleration due to gravity (-9.8 m/s²), and t is the time of flight.
Plugging in the values:
40 = V₀sin(θ) * t - 4.9t² -- (Equation 1)
Step 4: Calculate the horizontal displacement.
Given that the UFO is flying at a constant velocity of 70 m/s and passes directly over you, its horizontal displacement (range) will be the same as the projectile's horizontal displacement.
Using the horizontal motion equation:
Δx = V₀x * t,
where Δx is the horizontal displacement, V₀x is the initial horizontal velocity of the projectile (which will be V₀cos(θ)), and t is the time of flight.
Plugging in the values:
Δx = V₀cos(θ) * t -- (Equation 2)
Step 5: Eliminate time from the equations.
To solve for the lowest launch speed and launch angle, we need to eliminate time from Equations 1 and 2.
Rearrange Equation 2 to express t in terms of Δx:
t = Δx / (V₀cos(θ))
Substitute this expression for t in Equation 1:
40 = V₀sin(θ) * (Δx / (V₀cos(θ))) - 4.9 * (Δx / (V₀cos(θ)))²
Step 6: Find the launch speed and angle.
Rearrange the equation and simplify:
40 = (V₀sin(θ) * Δx) / (V₀cos(θ)) - 4.9 * (Δx²) / (V₀²cos²(θ))
Rewrite sin(θ) / cos(θ) as tan(θ):
40 = (V₀ * tan(θ) * Δx) - 4.9 * (Δx²) / (V₀²cos²(θ))
At this point, you can solve the equation using numerical methods or a graphing calculator to determine the lowest launch speed and angle.