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 find the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship, we need to consider the path of the projectiles fired by the enemy ship.
Given:
- Distance from the mountain peak to the enemy ship (d1) = 2153 m
- Height of the mountain peak (h) = 1690 m
- Initial speed of the projectiles fired by the enemy ship (vi) = 247 m/s
- Distance from the mountain peak to the western shoreline (d2) = 240 m
We can use the fact that projectiles follow a parabolic path to calculate the safe distances. The equation for the horizontal distance traveled by a projectile is given by:
d = vi * t,
where d is the horizontal distance, vi is the initial speed of the projectile, and t is the time taken by the projectile to reach the ground.
To calculate the time taken (t) by the projectile to reach the ground, we need to consider the vertical motion. The equation for the vertical motion of the projectile is given by:
h = (1/2) * g * t^2,
where h is the height of the mountain peak, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time taken.
Solving this equation for t, we get:
t = sqrt((2 * h) / g).
Now, we can substitute the value of t into the equation for horizontal distance (d) to get:
d = vi * t.
Let's calculate the safe distances:
1. Calculate the time taken (t):
t = sqrt((2 * h) / g)
= sqrt((2 * 1690) / 9.8)
≈ sqrt(344.69)
≈ 18.55 s (rounded to two decimal places)
2. Calculate the safe distance from the western shore:
- For the furthest safe distance:
d_furthest = vi * t
= 247 * 18.55
≈ 4578.85 m
≈ 4579 m (rounded to the nearest meter)
- For the closest safe distance, we need to consider the distance from the mountain peak to the western shoreline (d2):
d_closest = d2 - d_furthest
= 240 - 4579
= -4339 m
(Since the value is negative, it means that the safe distance is on the other side of the mountain peak. We can take the absolute value to get the positive distance.)
≈ 4339 m (rounded to the nearest meter)
Therefore, the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship are approximately 4339 meters and 4579 meters.