Calculating a Cyclist's Overall Displacement: A Step-by-Step Guide

To find the cyclist's overall displacement from the starting point, we can break down the journey into its components and then use the Pythagorean theorem to calculate the resultant displacement. Let's explore this step-by-step with the given problem:

Problem Statement

A cyclist travels 2.5 km east, 10 km south, and finally 5.0 km west. What is the cyclist's overall displacement from the starting point?

Step-by-Step Solution

Step 1: Break down the movements into components

Eastward movement: 2.5 km east Southward movement: 10 km south Westward movement: 5.0 km west

Step 2: Calculate the net east-west displacement

The net east-west displacement is:

East component: 2.5 km east (or 2.5 km) West component: 5.0 km west (or -5.0 km) Net East-West: 2.5 km - 5.0 km -2.5 km

This indicates that the cyclist travels 2.5 km west.

Step 3: Calculate the net north-south displacement

The net north-south displacement is:

North component: 0 km (no movement north) South component: 10 km south (or -10 km) Net North-South: 0 km - 10 km -10 km

This indicates that the cyclist travels 10 km south.

Step 4: Represent the displacements as vectors

We can represent the cyclist's displacement as a vector:

Displacement in the x-direction (east-west): -2.5 km (west) Displacement in the y-direction (north-south): -10 km (south)

Step 5: Calculate the overall displacement using the Pythagorean theorem

The overall displacement d can be calculated as follows:

d sqrt{x^2 y^2}

Substituting the values:

d sqrt{(-2.5)^2 (-10)^2} sqrt{6.25 100} sqrt{106.25} approx 10.31 km

Step 6: Determine the direction of the displacement

To find the direction angle ?θ of the displacement relative to the west direction, we can use the tangent function:

tanθ frac{opposite}{adjacent} frac{-10}{-2.5} 4

Calculating θ:

θ tan^{-1}(4) approx 75.96°

This angle is measured from the west towards the south.

Conclusion

The cyclist's overall displacement from the starting point is approximately 10.31 km at an angle of about 75.96° south of west.

Graphical Representation

The three displacements are combined using the component method. The x and y components of the three vectors are computed and added to obtain the sum of X and the sum of Y. The Pythagorean theorem is then used to solve for the resultant by using R^2 ΣX^2 ΣY^2. The direction of the resultant is computed by using the arc tangent function tan 1 (ΣX ÷ ΣY) and then measured from the X-axis counterclockwise.

The resultant is 10.31 km at 256°. The graph shows d_1 in blue line, d_2 in green line, and d_3 in magenta line while the resultant is in red line. The eastward direction is zero degrees, the southward direction is 270 degrees, and the westward direction is 180 degrees.