A ship, proceeding southward on a straight course at the rate of 12 miles/hr is, at noon,
40 miles due north of a second ship, which is sailing west at 15 miles/hr.
a) How fast are the ships approaching each other 1 hour later?
Let t hours be any time after noon
Make a sketch of that position.
Let the distance between them be D miles
I have a right-angled triangle and
D^2 = (15t)^2 + (40-12t)^2
2D dD/dt = 2(15t)(15) + 2(40-12t)(-12)
when t = 1, D^2 = 15^2 + 28^2
D = √1009
then when t = 1
dD/dt = .....
To determine how fast the ships are approaching each other, we can use the concept of relative velocity. Relative velocity is the difference in velocities between two objects.
We are given that the first ship is traveling south at 12 miles/hr and the second ship is traveling west at 15 miles/hr. We want to find their relative velocity or how fast they are approaching each other.
Step 1: Determine the velocity components of each ship.
- The first ship's velocity can be divided into two components: a southward component and an eastward component. Since the ship is moving strictly south, the eastward component is 0. Therefore, the southward component is 12 miles/hr.
- The second ship's velocity can be divided into two components: a westward component and a northward component. Since the ship is moving strictly west, the northward component is 0. Therefore, the westward component is 15 miles/hr.
Step 2: Calculate the relative velocity.
To calculate the relative velocity, we take the difference between the velocity components of the two ships. Since the northward component of the first ship and the eastward component of the second ship are both 0, we only need to consider the southward and westward components.
Relative velocity = southward component of first ship - westward component of second ship
Relative velocity = 12 miles/hr - 15 miles/hr
Relative velocity = -3 miles/hr
The negative sign indicates that the ships are approaching each other. Therefore, the ships are approaching each other at a speed of 3 miles per hour 1 hour later.