3. A man leaves his home by car and travels 5 km on a road running due East. The driver then
turns left and travels 2 km due north to a junction where he joins a road that goes North West
and drives a further 2 km. Find his displacement from home by then.
East:
5 - 2 /sqrt 2 = 3.5858
North:
2 + 2/sqrt 2 = 3.414
d^2 = 3.5858^2 + 3.414^2
d = 4.95 km
5km
To find the man's displacement from home, we need to calculate the straight-line distance between his current location and his starting point.
Let's break down the given information step by step:
1. The man travels 5 km due East. This implies he is 5 km to the right of his starting point.
2. He then turns left and travels 2 km due North. This implies he is 2 km above his current location (after traveling 5 km due East) and he is still 5 km to the right of his starting point.
3. At the junction, he joins a road that goes Northwest and drives a further 2 km. This means he is now 2 km in the Northwest direction from his current location (after traveling 2 km due North and 5 km due East).
Now, to find the man's displacement from home, we can use the Pythagorean theorem. According to the theorem, the square of the hypotenuse (the straight-line distance between two points) is equal to the sum of the squares of the other two sides (perpendicular to each other).
Using this theorem, we can calculate the displacement as follows:
Distance traveled East = 5 km
Distance traveled North = 2 km
Distance traveled Northwest = 2 km
To calculate the total displacement, we need to find the net eastward distance traveled and the net northward distance traveled.
Net eastward distance = Distance traveled East - Distance traveled Northwest
= 5 km - 2 km
= 3 km
Net northward distance = Distance traveled North
= 2 km
Now, let's use the Pythagorean theorem to find the displacement:
Displacement = √(Net eastward distance)^2 + (Net northward distance)^2
= √(3 km)^2 + (2 km)^2
= √(9 km^2 + 4 km^2)
= √(13 km^2)
= √13 km
≈ 3.61 km
Therefore, the man's displacement from home is approximately 3.61 km.