A ship maneuvers to within 2500 m of an island's 1800 m high mountain peak and fires a projectile at an enemy ship 610 m on the other side of the peak, as illustrated in Figure 3-29. If the ship shoots the projectile with an initial velocity of v = 256 m/s at an angle of θ = 75°, how close to the enemy ship does the projectile land? how close does the projectile come to the peak of the mountain?
To find out how close the projectile lands to the enemy ship and the peak of the mountain, we can break down the problem into two separate components: horizontal motion and vertical motion.
First, let's focus on the horizontal motion. The horizontal distance the projectile travels can be calculated using the horizontal component of its initial velocity. We can use the equation:
Horizontal distance = (Initial horizontal velocity) × (Time of flight)
The initial horizontal velocity can be found using the equation:
Initial horizontal velocity = Initial velocity × cos(θ)
where θ is the angle at which the projectile is launched.
Plugging in the given values, we have:
Initial horizontal velocity = 256 m/s × cos(75°)
Now, to determine the time of flight, we need to find the time it takes for the projectile to travel a horizontal distance of 610 m. We can use the equation:
Horizontal distance = (Initial horizontal velocity) × (Time of flight)
To isolate the time of flight, we rearrange the equation:
Time of flight = Horizontal distance / (Initial horizontal velocity)
Plugging in the values, we have:
Time of flight = 610 m / [(256 m/s) × cos(75°)]
Now that we have the time of flight, we can calculate the vertical distance the projectile travels. The vertical distance can be determined using the equation:
Vertical distance = (Initial vertical velocity) × (Time of flight) + (0.5) × (Acceleration due to gravity) × (Time of flight)^2
The initial vertical velocity can be found using the equation:
Initial vertical velocity = Initial velocity × sin(θ)
Applying the given values:
Initial vertical velocity = 256 m/s × sin(75°)
Now, we need to find the vertical distance the projectile travels. We can use the equation:
Vertical distance = (Initial vertical velocity) × (Time of flight) + (0.5) × (Acceleration due to gravity) × (Time of flight)^2
Where the acceleration due to gravity is approximately 9.8 m/s^2.
Plugging in the values:
Vertical distance = [(256 m/s) × sin(75°)] × (Time of flight) + (0.5) × (9.8 m/s^2) × (Time of flight)^2
Now we have the vertical distance. To find out how close the projectile lands to the enemy ship and the peak of the mountain, we have to calculate the horizontal and vertical components separately.
To find the distance from the enemy ship, we subtract the horizontal distance traveled by the projectile from the horizontal distance between the ship and the mountain:
Distance from enemy ship = 2500 m - Horizontal distance
To find the distance from the mountain peak, we subtract the vertical distance traveled by the projectile from the height of the mountain:
Distance from mountain peak = 1800 m - Vertical distance
Plugging in the calculated values will give you the final answers for both distances.