Question

A particle undergoes three successive displacements in a plane, as follows: \(4.13 \mathrm{~m}\) southwest, \(5.26 \mathrm{~m}\) east, and \(5.94 \mathrm{~m}\) in a direction \(64.0^{\circ}\) north of east. Choose the \(x\) axis pointing east and the \(y\) axis pointing north and find \((a)\) the components of each displacement, \((b)\) the components of the resultant displacement, \((c)\) the magnitude and direction of the resultant displacement, and \((d)\) the displacement that would be required to bring the particle back to the starting point.

Short answer

The components of each displacement are Displacement 1: (-4.13m, -4.13m), Displacement 2: (5.26m, 0m), Displacement 3: \(5.94m \cdot cos(64)\,m, 5.94m \cdot sin(64)\,m\). The components of the resultant displacement are given by the sum of the respective x and y components from each displacement. The magnitude of the resultant displacement is calculated using Pythagoras' theorem and the direction is found using the tangent of the ratio of the y and x components. The displacement required to return to the start is just the negative of these resultant components.

Step-by-step solution

Calculate components for each displacement

Firstly, calculate the components for each displacement, bearing in mind that in the southwest direction, both x and y components are negative.\r\nFor displacement 1 (4.13m southwest), both x and y components are same: \(-4.13 m\).\r\nFor displacement 2 (5.26m east), \(x = 5.26m\) and \(y = 0m\).\r\nFor displacement 3 (5.94m, 64 degrees north of east), \(x = 5.94m \cdot cos(64^\circ)\) and \(y = 5.94m \cdot sin(64^\circ)\).

Calculate components of the resultant displacement

The resultant displacement can be calculated by simply adding the respective x and y components from each displacement. The x components are \(-4.13m\), \(5.26m\), \(5.94m \cdot cos(64^\circ)\). The y components are \(-4.13m\), \(0m\), \(5.94m \cdot sin(64^\circ)\). So the resultant displacement in the x direction is \(-4.13m + 5.26m + 5.94m \cdot cos(64^\circ)\) and in the y direction is \(-4.13m + 0m + 5.94m \cdot sin(64^\circ)\).

Calculate magnitude and direction of the resultant displacement

The magnitude of the resultant displacement will be \(\sqrt{{(Resultant\_x)}^2 + {(Resultant\_y)}^2}\) and direction can be calculated as atan(\(Resultant\_y / Resultant\_x)\) if Resultant_x > 0 else atan(\(Resultant\_y / Resultant\_x)\) + 180. This is computed in degrees.

Calculate displacement required to return to the starting point

The displacement needed to return to the start, or 'the return vector', is just the negative of the resultant vector. So, calculate \(-Resultant_x\) and \(-Resultant_y\), which will be the x and y component of the displacement that brings the particle back to the starting point.

Key concepts

Displacement Calculation

When dealing with problems involving movement in a plane, calculating displacement is key. Displacement is a vector quantity, meaning it has both magnitude and direction. In our scenario, we are tasked with finding the total displacement of a particle undergoing three separate movements. Each movement has a certain magnitude and direction, and we must account for these to solve for total displacement.

To compute the displacement for each step, we must first break them down into their components. This means identifying the movement along the x (east-west) and y (north-south) axes. This task involves:

• Identifying directions: Displacement toward the east has a positive x component, and toward the north has a positive y component. Southwest means both x and y components are negative.
• Breaking down vectors: We'll use trigonometric functions such as cosine and sine to find each component when angles are involved, like in the third displacement.

Understanding these components helps us to add them together correctly in the next steps.

Resultant Vector

Once the displacements are separated into their components, the next step is to find the resultant vector. This resultant vector represents the overall effect of the multiple displacements on the particle's position.

To find the components of the resultant vector, simply add the x-components of all displacements together to find the resultant x-component. Similarly, add up all the y-components for the resultant y-component:

• Sum of x-components: This is the combined effect of all east-west movements.
• Sum of y-components: This gives the overall north-south movement.

By analyzing these collected sums, we get the complete picture of how the particle's position has shifted overall. This vector then serves as the basis to find magnitude and direction.

Coordinate System

A consistent coordinate system is essential for solving problems involving vectors, especially in a two-dimensional plane. In this exercise, the coordinate system is defined with the x-axis pointing east and the y-axis pointing north.

Choosing an appropriate coordinate system helps in:

• Standardizing calculations: All displacements are described in terms of the same directionality. For example, an eastward displacement only affects the x-axis.
• Using trigonometry with ease: Angles and trigonometric functions assume a knowledge of which direction each axis points.

This uniformity bypasses the confusion that could arise from an arbitrary or inconsistent system, allowing better focus on accurately calculating results.

Trigonometry in Physics

Trigonometry is a powerful tool in physics, especially for problems involving vectors and directions. It allows us to break down complex vector quantities into comprehensible components using angles.

For the displacements described in the exercise, trigonometric functions like cosine and sine become crucial, particularly for the third displacement where an angle is involved:

• Cosine function (\(\cos\)) helps find the component along the x-axis, especially for angles measured from the horizontal (east).
• Sine function (\(\sin\)) provides the component parallel to the y-axis, as it relates to how much the vector leans towards vertical (north-south).

By leveraging these functions, we systematically decompose movement into measurable components, allowing for straightforward addition of these vectors. This process highlights trigonometry's vital role in applications beyond simple math, serving as a bridge to understand physical changes in movement.