A car travles 6km towards north at an angle of `45^(@)` to the east and then travles distance of 4km towards north at an angle of `135^(@)` to east (figure). How far is the point from the starting point? What angle does the straight line joining its initial and final position makes with the east?

Options

A. `sqrt50 km and tan^(-1) (5)`
B. `10 km and tan^(-1) (sqrt5)`
C. `sqrt52 km and tan^(-1) (5)`
D. `sqrt52km and tan^(-1) (sqrt5)`

Answer

C

Explanation

Displacement in x direction
`S_(x)= (6 cos 45^(@)- 4 cos 45) hat(i)`
`= (2 xx (1)/(sqrt2)) hat(i)= (sqrt2 km) hat(i)`
Displacement in y-direction
`S_(y)= (6 sin 45^(@) + 4 sin 45^(@)) hat(j)`
`= 10 xx (1)/(sqrt2) hat(j)= (5 sqrt2 km) hat(j)`
Net displacement
`S= sqrt(S_(x)^(2) + S_(y)^(2))= sqrt((sqrt2)^(2) + (5 sqrt2)^(2))= sqrt52km`
Angle makes with East direction `tan theta= (S_(y))/(S_(x))= (5 sqrt2)/(sqrt2)`
`rArr theta= tan^(-1) (5)`