Answer:
Aprox: 465.4 ms/2
Explanation:
To find the magnitude of the acceleration the engine must produce, we'll break down the missile's motion into its horizontal and vertical components.
Given:
Initial velocity (V₀) = 1810 m/s
Launch angle (θ) = 20.0°
Target distance (d) = 19,500 m
Target direction (φ) = 32.0°
Time of flight (t) = 9.20 s
First, let's find the horizontal and vertical components of the missile's initial velocity (V₀x and V₀y).
V₀x = V₀ * cos(θ)
V₀y = V₀ * sin(θ)
V₀x = 1810 m/s * cos(20.0°) ≈ 1712.4 m/s
V₀y = 1810 m/s * sin(20.0°) ≈ 618.8 m/s
Now, let's find the horizontal and vertical displacements (dx and dy) of the missile using the time of flight (t) and the target distance (d).
dx = d * cos(φ)
dy = d * sin(φ)
dx = 19,500 m * cos(32.0°) ≈ 16,402.8 m
dy = 19,500 m * sin(32.0°) ≈ 10,236.8 m
Next, let's calculate the acceleration needed in the x-direction (ax) and y-direction (ay) to cover the horizontal and vertical displacements in the given time.
ax = (2 * dx) / t²
ay = (2 * dy) / t²
ax = (2 * 16,402.8 m) / (9.20 s)² ≈ 391.7 m/s²
ay = (2 * 10,236.8 m) / (9.20 s)² ≈ 257.7 m/s²
Finally, to find the magnitude of the acceleration, we can use the Pythagorean theorem:
acceleration magnitude (a) = √(ax² + ay²)
a = √(391.7 m/s²)² + (257.7 m/s²)² ≈ 465.4 m/s²
Therefore, the magnitude of the acceleration that the engine must produce is approximately 465.4 m/s².