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.