An enemy ship is on the east side of a mountain island, as shown in the figure. The enemy ship has maneuvered to within d1 = 2153 m of the h = 1690 m high mountain peak and can shoot projectiles with an initial speed of vi = 247 m/s. If the western shoreline is horizontally d2 = 240 m from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship?
To solve this problem, we need to calculate the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship.
First, let's set up a figure for better visualization. Imagine a right triangle with the mountain peak as the top vertex, the enemy ship as the right vertex, and the western shoreline as the left vertex. The distance from the mountain peak to the enemy ship is d1 = 2153 m, and the height of the mountain peak is h = 1690 m.
We can use trigonometry and projectile motion equations to find the safe distances from the western shore. Let's break down the problem into steps:
Step 1: Determine the angle between the line connecting the mountain peak and the enemy ship and the horizontal axis. We can use the tangent function:
tan(θ) = opposite/adjacent = h/d1
θ = arctan(h/d1)
Step 2: Calculate the time it takes for a projectile to reach the peak height of the mountain. Use the kinematic equation:
h = (1/2) * g * t^2
t = sqrt(2h/g)
where g is the acceleration due to gravity (~9.8 m/s^2).
Step 3: Find the horizontal distance covered by the projectile at this time. Use the horizontal velocity component:
d_x = v_x * t
v_x = vi * cos(θ)
Step 4: Calculate the total distance from the western shore where a ship can be safe by subtracting the horizontal distance from the distance to the mountain peak:
d_safe = d2 - d_x
Now we have the value of d_safe, which represents the distance from the western shore at which a ship can be safe from the bombardment of the enemy ship.
Plug in the given values and follow the steps to find the solution:
Given:
d1 = 2153 m (distance from the mountain peak to the enemy ship)
h = 1690 m (height of the mountain peak)
vi = 247 m/s (initial speed of the projectiles)
Step 1: Calculate θ
θ = arctan(h/d1)
θ = arctan(1690/2153)
θ ≈ 39.94°
Step 2: Calculate t
t = sqrt(2h/g)
t = sqrt(2 * 1690 / 9.8)
t ≈ 18.26 s
Step 3: Calculate d_x
v_x = vi * cos(θ)
v_x = 247 * cos(39.94°)
v_x ≈ 189.04 m/s
d_x = v_x * t
d_x = 189.04 * 18.26
d_x ≈ 3455.07 m
Step 4: Calculate d_safe
d_safe = d2 - d_x
d2 = 240 m (distance from the mountain peak to the western shoreline)
d_safe = 240 - 3455.07
d_safe ≈ -3215.07 m
The distance d_safe comes out to be negative, which is not physically meaningful in this context. Therefore, there is no distance from the western shore where a ship can be safe from the bombardment of the enemy ship.