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 = 248 m/s at an angle of θ = 74°, how close to the enemy ship does the projectile land?

Question

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 = 248 m/s at an angle of θ = 74°, how close to the enemy ship does the projectile land?

Answer 1

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Answer 2

To solve this problem, we can use the principles of projectile motion. The projectile's motion can be split into two components: horizontal and vertical.

First, let's calculate the time it takes for the projectile to reach the peak of the mountain. We can use the vertical component to determine this. We know the initial vertical velocity (v₀y) is given by v₀y = v₀ * sin(θ), where v₀ is the initial velocity and θ is the launch angle. In this case, v₀y = 248 m/s * sin(74°).

Using the equation of motion: Y = Y₀ + v₀y * t - (1/2) * g * t², where Y represents the displacement and Y₀ is the initial vertical position, we can plug in the known values: Y = 1800 m, Y₀ = 0 m (assuming the ground level is at 0 m), and g = 9.8 m/s² (acceleration due to gravity). We can then solve for t, which represents the time taken to reach the peak of the mountain.

Once we have the time taken to reach the peak, we can calculate how far the projectile travels horizontally during that time. We know the initial horizontal velocity (v₀x) is given by v₀x = v₀ * cos(θ), where v₀ is the initial velocity and θ is the launch angle. In this case, v₀x = 248 m/s * cos(74°).

The horizontal distance traveled during time t is given by X = v₀x * t.

Now, we need to calculate the time it takes for the projectile to travel from the peak of the mountain to the enemy ship. We can use the known horizontal distance (610 m) and the horizontal velocity (v₀x) to determine this. Since the horizontal velocity remains constant, the time of flight (T) is given by T = distance / v₀x.

Finally, we can calculate the horizontal distance (X) traveled during time T using the equation X = v₀x * T, which will give us the distance from the peak of the mountain to where the projectile lands.

By following these steps and plugging in the given values, we can find the final answer of how close to the enemy ship the projectile lands.