Please Help ASAP! Motion in two dimensions!

Question

A Coast Guard cutter detects an unidentified ship at a distance 35.0 km in the direction 25.0° east of north. The ship is traveling at 46.0 km/h on a course at 35.0° east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel.

(a) If the speedboat travels at 50.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north.

(b) Find the time required for the cutter to intercept the ship.

Answer 1

Hi Kaylyn. Relax! This is easier than it appears.

Let's denote the ship by 1 and the speedboat by 2.

The question doesn't specifically state that neither boat is accelerating so we can assume that.

Now we look at their equations of motion in the horizontal (east/west) and vertical (north/south) directions, separately. We can do that because these 2 directions are orthogonal (at right angles) and so can't affect each other.)

For the ship, x₁(t) = (46 sin35^o) t

and y₁(t) = (46 cos35^o) t

Suppose the speedboat makes angle θ with the vertical (E of N)

Then, for the speedboat, x₂(t) = (50 sinθ) t

and y₂(t) = (50 cosθ) t

When the speedboat intercepts the ship, their respective x and y co-ordinates will be the same.

So, (50 cosθ) t = (46 cos35^o) t

=> cosθ = 46 cos35^o / 50

We can now solve for θ. I leave this as an exercise for you.

Part (b) The only information we haven't used is the initial 35 km distance from the origin to the ship.

We use that to find the time taken as follows. We can calculate the x-coord of the ship's initial position. Then, if they meet in time t, the speedboat must go that distance further in the same time t.

I leave that as an exercise for your mind!