When analyzing the movement of an object that changes direction multiple times, one critical aspect is determining the total displacement. This article explores a specific scenario where a body first moves 10 meters towards the north, then 15 meters towards the east, and finally 20 meters vertically upwards. We will break down the movement into components and compute the total displacement using vector addition and the Euclidean distance formula.

Understanding the Movement Components

The movement of the body can be described in three components: north, east, and vertically upwards. Each movement can be represented as a vector in three-dimensional space:

Northward Movement

First, the body moves 10 meters towards the north. We can represent this movement as the vector A:

0, 10, 0

Eastward Movement

Next, the body moves 15 meters towards the east. This movement can be represented as the vector B:

15, 0, 0

Upwards Movement

Finally, the body moves 20 meters vertically upwards. This movement can be represented as the vector C:

0, 0, 20

Calculating the Total Displacement Vector

The total displacement vector D is the sum of the individual displacement vectors A, B, and C:

Total Displacement

15, 10, 20

Using the vector addition, the total displacement can be calculated as:

Magnitude of the Displacement Vector

The magnitude of the displacement vector can be calculated using the Euclidean distance formula in three-dimensional space:

[ lVert mathbf{D} rVert sqrt{x^2 y^2 z^2} ]

Where:

( x 15 ) meters ( y 10 ) meters ( z 20 ) meters

Solving for (lVert mathbf{D} rVert):

[ lVert mathbf{D} rVert sqrt{15^2 10^2 20^2} ] [ lVert mathbf{D} rVert sqrt{225 100 400} ] [ lVert mathbf{D} rVert sqrt{725} ] [ lVert mathbf{D} rVert approx 26.93 , text{meters} ]

Therefore, the total displacement of the body is approximately 26.93 meters.

Coordinate Representation

After the three movements, the body is at a point ( P ). Its coordinates can be represented as:

X-coordinate: 15 meters (towards the North) Y-coordinate: 20 meters (towards the East) Z-coordinate: 10 meters (vertically upwards)

Let the origin be ( O ). The total displacement of the body is the distance ( OP ), which can be calculated using the Euclidean distance formula:

[ OP sqrt{x^2 y^2 z^2} ] [ OP sqrt{15^2 20^2 10^2} ] [ OP sqrt{225 400 100} ] [ OP sqrt{725} ] [ OP approx 26.926 , text{meters} ]

Rounding to 2 significant digits, the total displacement is approximately 27 meters.

Pythagorean Theorem Application

The total displacement can also be determined using the Pythagorean theorem in two parts:

The root of (10 , text{m}^2 15 , text{m}^2) gives the 2D hypotenuse in the horizontal plane. The found hypotenuse and the vertical translation are then used with the Pythagorean theorem to find the total distance from the origin.

The total displacement can be mathematically expressed as:

[ text{Total displacement} sqrt{sqrt{10^2 15^2} 20^2} ] [ text{Total displacement} sqrt{sqrt{100 225} 400} ] [ text{Total displacement} sqrt{sqrt{325} 400} ] [ text{Total displacement} sqrt{18 400} ] [ text{Total displacement} approx 26.93 , text{meters} ]

In conclusion, the total displacement of the body is approximately 26.93 meters, taking into account both horizontal and vertical distances.