- #1

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- Homework Statement:
- Two ships A and B begin at the same time from their respective points 4 kms apart along the ##y## axis with speeds 3 km/h and 1 km/h (uniformly) respectively, as shown in the figure below. The motion of ship B is parallel to the ##x## axis while that of A is at an angle of ##60^{\circ}## to the ##x## axis. ##\textbf{Will the two ships collide at point P}?##

- Relevant Equations:
- For unaccelerated motion, distance travelled ##d = v_0 t##, where ##v_0## is the uniform speed. Also, in a right angled triangle, ##\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}##.

There are two ways to solve. My problem is, when I apply the two, I end up with a

**contradiction**.

**Method 1 :**Conclusion - Ships will

__not collide__.

Imagine the ships collide after a time ##t##.

Since they both start at the same time, the distances travelled by them would be ##3t## (A) and ##t## (B), respectively. (I have shown the two distances in the diagram, in terms of ##t##).

Using parallel-line and angles, we see that ##\angle BPA = 60^{\circ}##. Thus if we take the cosine of the angle using trigonemetrical ratios of a right angled triangle, we have ##\cos 60^{\circ} = \frac{t}{3t} = \frac{1}{3}##!

We know that ##\cos 60^{\circ} = \frac{1}{2}##, hence it's clear that two two ships will

__not collide__.

**Method 2 :**Conclusion - Ships will

__collide__.

Imagine the ships collide after a time ##t##.

Since they both start at the same time, the distances travelled by them would be ##3t## (A) and ##t## (B), respectively. (I have shown the two distances in the diagram, in terms of ##t##).

In the right angled ##\triangle ABP##, using the Pythagorean theorem, we have ##(3t)^2 = t^2 + 4^2 \Rightarrow 9t^2 = t^2 + 16 \Rightarrow 8t^2 = 16 \Rightarrow t^2 = 2 \Rightarrow t = \sqrt{2}\; \text{hr} \approx 85 \text{minutes}## .

Hence the two ships

__will collide__after a time of 85 minutes from start.

**Contradiction**

*One possible way out is to note that nowhere in my second solution did I consider the fact that Ship A was moving at an angle of 60 degrees to the horizontal. What I*##\cos^{-1} \frac{t}{3t} = \cos^{-1} \frac{1}{3} = 70.5^{\circ}##.

__did__show was that with the speeds as given, the two ships can collide after the given time (85 minutes). If that was to happen, then the angle at which Ship A should move would be*It becomes a different problem really : Given the speeds of 1 and 3 km/h, calculate the angle at which ship A should move in order that they collide.*

Am I right? When I drew the relative velocity of B with respect to A (##\vec v_{BA}##), I found that it was pointing away from A, clearly showing that ship B would not be moving in A's direction and that

__no collision__would take place.