Ship A is sailing due west at 8km/hr when it sees ship B northwest of its present position. If ship B is sailing due south at 3km/hr, then how close will they come on their courses?
It depends on how far away B is when sighted. If it is k km north and west, then the distance z between the ships after t hours is
z^2 = (k-3t)^2 + (k-8t)^2
= 73t^2-22kt+k^2
dz/dt = (73t - 11k)/√(73t^2-22kt+k^2)
dz/dt=0 when t = 11k/73
so, the minimum distance is
z = 5k/√73
To find out how close ship A and ship B will come on their courses, we can use vector addition.
Let's break down the velocities of ship A and ship B into their components:
Velocity of ship A:
- In the x-axis (west-east direction): 8 km/hr
- In the y-axis (north-south direction): 0 km/hr (since it is traveling strictly west)
Velocity of ship B:
- In the x-axis (west-east direction): 0 km/hr (since it is traveling strictly south)
- In the y-axis (north-south direction): -3 km/hr (negative because it is traveling south)
Now, we can add the velocities of ship A and ship B to get the total velocity:
Total velocity in the x-axis: 8 km/hr + 0 km/hr = 8 km/hr
Total velocity in the y-axis: 0 km/hr + (-3 km/hr) = -3 km/hr
Now, we can use the Pythagorean theorem to find the magnitude of the total velocity:
Total velocity magnitude = sqrt((8 km/hr)^2 + (-3 km/hr)^2)
= sqrt(64 km^2/hr^2 + 9 km^2/hr^2)
= sqrt(73 km^2/hr^2)
≈ 8.544 km/hr
The magnitude of the total velocity represents the speed at which the two ships are approaching each other on their courses. Therefore, the two ships will come approximately 8.544 km closer together every hour.
To determine how close the two ships will come on their courses, we can use the concept of relative velocity. Relative velocity refers to the velocity of an object in relation to another object.
In this scenario, ship A is sailing due west at 8 km/hr, while ship B is sailing due south at 3 km/hr. To find their relative velocity, we need to break down their velocities into their respective components.
The velocity of ship A can be represented as follows:
- Horizontal component (V_Ah) = 8 km/hr (westward)
- Vertical component (V_Av) = 0 km/hr (no north/south motion)
The velocity of ship B can be represented as follows:
- Horizontal component (V_Bh) = 0 km/hr (no east/west motion)
- Vertical component (V_Bv) = 3 km/hr (southward)
Next, we need to find the resultant velocity by adding the horizontal and vertical components of ship A and ship B separately.
The resultant velocity components can be calculated as follows:
- Resultant horizontal component = V_Ah + V_Bh
- Resultant vertical component = V_Av + V_Bv
Resultant horizontal component = 8 km/hr + 0 km/hr = 8 km/hr (westward)
Resultant vertical component = 0 km/hr + (-3) km/hr = (-3) km/hr (northward)
Now that we have the resultant velocity (8 km/hr westward, and -3 km/hr northward), we can find the magnitude (speed) of the relative velocity using the Pythagorean theorem:
Relative velocity (V_R) = √[(Resultant horizontal component)^2 + (Resultant vertical component)^2]
Relative velocity (V_R) = √[(8 km/hr)^2 + (-3 km/hr)^2]
Relative velocity (V_R) = √[64 km^2/hr^2 + 9 km^2/hr^2]
Relative velocity (V_R) = √73 km^2/hr^2
Relative velocity (V_R) = √73 km/hr
Hence, the relative velocity between the ships is √73 km/hr.
To determine how close the two ships will come on their courses, we consider the time it takes for them to cross paths. Since their paths are perpendicular to each other, the point at which they will be closest will occur at the midpoint of their paths.
Using the formula: Distance = Relative velocity × Time
We need to find the time it takes for the ships to cross paths.
Time = Distance / Relative velocity
Since Ship A is sailing at a constant speed of 8 km/hr, the time taken to cross paths can be calculated using the horizontal distance traveled (ship A's velocity component) and the relative velocity:
Time = Horizontal distance (ship A) / Relative velocity
Time = Horizontal distance (ship A) / √73 km/hr
To determine the horizontal distance traveled by ship A, we need to know how long it takes for ship A and ship B to cross paths. Since the distance is not given, we cannot determine the exact closest point of approach without more information.