When dealing with displacement in physics, it is important to understand how to calculate the total displacement of a particle. A common misconception is to simply add the magnitudes of displacements without considering the direction. In this context, we will explore the statement: ldquo;If a particle has a displacement of 10 meters followed by a displacement of 30 meters, then the total displacement will be 40 meters.rdquo; Is this true or false, and why?
Directional Consistency
The statement is correct only if the particle moves in the same direction for both displacements. If the direction of the second displacement is different from the first, the statement is false. To understand why, letrsquo;s break down the scenario in detail:
Scenario 1: Same Direction
Example: If a particle first moves 10 meters east, and then moves another 30 meters east, the total displacement is indeed 40 meters east. This calculation is straightforward because the displacements are in the same direction. The vector addition of these displacements is:
10 meters 30 meters 40 meters
Scenario 2: Different Directions
Example: If a particle first moves 10 meters east, and then moves 30 meters north, the net displacement is not simply 40 meters. Instead, the net displacement is the diagonal of a rectangle with sides 10 meters and 30 meters. Using the Pythagorean theorem, the net displacement can be calculated as follows:
Net displacement √(102 302) √(100 900) √1000 ≈ 31.62 meters
The direction of this net displacement can be found using trigonometry. The angle made with the east direction (the reference direction in this case) can be calculated using the tangent function:
tan(θ) 30 / 10 3
θ arctan(3) ≈ 71.57°
Thus, the particle will be 31.62 meters away from the origin, at an angle of approximately 71.57° from the east.
Vector Addition in Displacement
The principle of vector addition is central to understanding displacement. Displacement is a vector quantity, which means it has both magnitude and direction. When adding two or more displacements, you must consider the direction of each displacement. Vector addition can be done graphically or using algebraic methods such as the Pythagorean theorem or the cosine law for non-perpendicular vectors.
Graphical Method
A graphical method involves drawing the displacements as vectors and using the head-to-tail method to find the resultant vector, which represents the total displacement. In the case of 10 meters east and 30 meters north, you would draw a 10-meter vector pointing east, and a 30-meter vector starting from the head of the first vector, pointing north. The line connecting the tail of the first vector to the head of the last vector (30-meter north vector) represents the net displacement.
Algebraic Method
The algebraic method uses trigonometric functions or the Pythagorean theorem to find the magnitude and direction of the resultant displacement. For a 10-meter east and 30-meter north displacement,
Magnitude:
Net displacement √(102 302) √1000 ≈ 31.62 meters
Direction:
θ arctan(30 / 10) arctan(3) ≈ 71.57°
So, even if a particle displaces 10 meters in one direction and 30 meters in another, the total displacement is not the simple sum of the magnitudes unless the directions are the same.
Misconceptions: Directional Dependence
A common misconception is that one can simply add the magnitudes of displacements regardless of direction. This is only true if the directions are aligned. If they are not, the direction of the resultant displacement must be taken into account. As demonstrated in the graphical and algebraic methods, the net displacement can vary significantly based on the orientation of the individual displacements.
For instance, if a particle starts at the origin and moves 10 meters east and then 30 meters north, the total displacement is 31.62 meters in a direction 71.57° north of east. This is not the same as a simple addition of 40 meters in a single direction.
Conclusion
In summary, the statement ldquo;If a particle has a displacement of 10 meters followed by a displacement of 30 meters, then the total displacement will be 40 metersrdquo; is true only if the movements are in the same direction. If the directions are different, the total displacement can be any value between 20 meters (when moving in opposite directions) and 40 meters (when moving in the same direction), depending on the angle between the displacements.
Understanding vector addition and the importance of direction in displacement is crucial for accurate calculations in physics. Whether using graphical or algebraic methods, the direction of each displacement should always be considered when determining the total displacement.