Relative motion involving vectors(kinematics)

[Jharr94]

Homework Statement

Ship A is located 3.5 km north and 2.8 km east of ship B. Ship A has a velocity of 22 km/h toward the south and ship B has a velocity of 40 km/h in a direction 37° north of east.

At what time is the separation between the ships least?

Homework Equations

V_ab=V_b-V_a
Where V_ab is the velocity of a with respect to b and V_a or V_b is the velocity of that ship.

The Attempt at a Solution

I'm thinking this part of the problem involves relative motion as well as kinematics, however I can't find out how to get the LEAST distance, the only thing I can think is when the positions are equal, which they should never be however,

I've found the position vector of a with respect to b as:

(2.8-31.95t)i+(3.5-46.07t)j=r_fab (r_fab is final position of a with respect to b)

from this I'm stuck as to where to go, I don't know how to find the LEAST distance between the two, I tried setting the position equations equal to each other( r_b=r_ab) but I can't seem to find answer because there are two different times for the x component and the y component.

Help please, thanks.

 

[HallsofIvy]

If the relative position vector is (a+ bt)i+ (c+ dt)j, then the distance between them is $\sqrt{(a+bt)^2+ (c+ dt)^2}$. Do you know how to minimize that?

 

[Architeuthis Dux]

Great job on using the correct equations for relative motion involving vectors. To find the least distance between the ships, you will need to use the Pythagorean theorem to find the magnitude of the relative position vector (r_fab). This will give you the distance between the two ships at any given time.

To find the time at which the separation between the ships is the least, you will need to differentiate the magnitude of r_fab with respect to time and set it equal to zero. This will give you the time at which the separation is at a minimum.

Alternatively, you can also use the concept of minimum and maximum points on a graph to determine the time at which the separation is the least. Plot the magnitude of r_fab as a function of time and find the point on the graph where the slope is zero. This will also give you the time at which the separation is at a minimum.

I hope this helps you solve the problem. Keep up the good work!