Question

To go from the student hostels to the university library, a student must cover 650m in a direction 47^o south of west. To go from the football ground to the library, the student must move straight south for 420m. How far and in what direction must the student move to go from hostels to the football ground?

Answer 1

As we know from the condition of the question, to go from the student hostels to the university library, a student must cover 650 m in a direction $47^{\circ}$south of west. To go from the football ground to the library, the student must move straight south for 420 m. So, if the student will go from the library straight north for 420 m, he will go to the football ground. Then, his resultant displacement will be the displacement that he must move to go from hostels to the football ground. Let's find the resultant displacement of the student:

$$d_x=650\ m\cdot cos(180^{\circ}+47^{\circ})+420\ m\cdot cos90^{\circ}=-443.3\ m,$$ $$d_y=650\ m\cdot sin(180^{\circ}+47^{\circ})+420\ m\cdot sin90^{\circ}=-55.38\ m.$$

Then, the magnitude of the resultant displacement can be found from the Pythagorean theorem:

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(-443.3\ m)^2+(-55.38\ m)^2}=446.74\ m.$$

We can find the angle as follows:

$$\theta=sin^{-1}(\dfrac{d_y}{d})=sin^{-1}(\dfrac{-55.38\ m}{446.74\ m})=-7.12^{\circ}.$$

The sign minus means that the resultant displacement has direction $7.12^{\circ} S\ of\ W$.

Therefore, the student must move 446.74 m in a direction of $7.12^{\circ} S\ of\ W$ to go from hostels to the football ground.