A man goes 24 m towards east and then 10 m towards north. How far is he away from his initial position?
26 m
Solving Displacement Using the Pythagorean Theorem
This question asks us to find the distance of a man from his initial position after he moves in two perpendicular directions: east and then north. This scenario can be visualized as forming a right-angled triangle, where the movements east and north are the two shorter sides (legs), and the distance from the starting point is the longest side (hypotenuse).
The man's movement can be broken down into two parts:
- Movement 1: 24 m towards the east.
- Movement 2: 10 m towards the north.
Since east and north directions are perpendicular to each other, the two displacements are at a 90-degree angle. The distance from the initial position is the magnitude of the resultant displacement vector, which is the hypotenuse of the right triangle formed by these movements.
We can use the Pythagorean theorem to find the length of the hypotenuse (the distance from the initial position). The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this is expressed as:
\[ c^2 = a^2 + b^2 \]
In this case:
- Side a = Displacement towards east = 24 m
- Side b = Displacement towards north = 10 m
- Hypotenuse c = Distance from the initial position (what we need to find)
Step-by-Step Calculation of Distance
Let 'd' be the distance from the initial position.
According to the Pythagorean theorem:
\[ d^2 = (\text{East displacement})^2 + (\text{North displacement})^2 \]
Substitute the given values:
\[ d^2 = (24 \, \text{m})^2 + (10 \, \text{m})^2 \]
Calculate the squares:
\[ d^2 = 576 \, \text{m}^2 + 100 \, \text{m}^2 \]
Add the squared values:
\[ d^2 = 676 \, \text{m}^2 \]
To find 'd', take the square root of both sides:
\[ d = \sqrt{676 \, \text{m}^2} \]
Calculate the square root:
\[ d = 26 \, \text{m} \]
So, the man is 26 meters away from his initial position.
Summary of Displacement Calculation
| Movement Direction | Distance (m) |
|---|---|
| East | 24 |
| North | 10 |
| Calculation Step | Value |
|---|---|
| \(24^2\) | 576 |
| \(10^2\) | 100 |
| \(24^2 + 10^2\) | \(576 + 100 = 676\) |
| \(\sqrt{676}\) | 26 |
The final distance from the initial position is 26 m.
Revision Table: Displacement and Distance Concepts
| Concept | Definition | Nature | Example in this problem |
|---|---|---|---|
| Distance | The total path length covered. | Scalar (magnitude only) | 24 m + 10 m = 34 m (total path) |
| Displacement | The shortest distance from the initial to the final position. Includes direction. | Vector (magnitude and direction) | 26 m at an angle North of East from start |
Additional Information: Vectors in Physics
In physics, quantities like displacement, velocity, and force are vectors because they have both magnitude (size) and direction. Distance and speed are scalar quantities because they only have magnitude.
When dealing with vector displacements that are perpendicular, we can use the Pythagorean theorem to find the magnitude of the resultant displacement. If the displacements are not perpendicular, vector addition methods like the triangle rule or parallelogram rule are used, often involving trigonometry.
Understanding the difference between distance and displacement is crucial in physics problems involving motion.
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