Question

Consult your online resources for additional practice. A man walks \(30 \mathrm{m}\) east and then \(40 \mathrm{m}\) north. What is the difference between his traveled distance and his displacement? (A) 0 m (B) 20 m (C) 50 m (D) 70 m

Short answer

The difference is 20 m.

Step-by-step solution

Understanding Traveled Distance

Traveled distance is the total length of the path the man walked. So, add the distance walked east (30 m) and the distance walked north (40 m).

Calculate the Total Traveled Distance

Total traveled distance = 30 m (east) + 40 m (north) = 70 m.

Understanding Displacement

Displacement is the straight line distance from the starting point to the ending point. It can be found using the Pythagorean theorem for a right triangle.

Calculate the Displacement

Use the Pythagorean theorem to calculate the displacement. \[ \text{Displacement} = \sqrt{(30 \mathrm{m})^2 + (40 \mathrm{m})^2} = \sqrt{900 + 1600} = 50 \mathrm{m} \].

Find the Difference

Subtract the displacement from the traveled distance to find the difference: 70 m (traveled distance) - 50 m (displacement) = 20 m.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is an important concept in mathematics. It helps you find the lengths of the sides in a right triangle. According to the theorem, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as:

\(c^2 = a^2 + b^2\).

To use this formula, you need to know two sides of the triangle. In the exercise, the man walks 30 meters east and 40 meters north. These two distances form the two sides of a right triangle. Using the Pythagorean theorem, we can find the straight-line distance from the starting point to the ending point, which is known as displacement.

Kinematics

Kinematics is the study of motion, involving calculations of distance and displacement, speed, velocity, and acceleration. In the context of our exercise, we are focused on distance and displacement.

Distance refers to the total path length covered, without regard to direction. On the other hand, displacement is the shortest straight-line distance from the initial to the final position, accounting for direction.

In kinematics, understanding these differences is essential. It helps you solve problems involving daily activities like walking, driving, or moving objects. The exercise demonstrates how to apply these concepts practically by calculating both the total distance traveled and the straight-line displacement.

Distance and Displacement

Distance and displacement might sound similar, but they are different.

Distance is a scalar quantity. It only considers how much ground is covered, not the direction. For example, if you walk 30 meters east and then 40 meters north, the total distance you've walked is 70 meters.

Displacement is a vector quantity. It considers both distance and direction. It is the shortest path from the starting point to the ending point. In our exercise, we can calculate the displacement using the Pythagorean theorem, as the two movements (30 meters east and 40 meters north) form a right triangle. \[ \text{Displacement} = \sqrt{(30m)^2 + (40m)^2} = 50m \].

So, while distance covered is 70 meters, the actual displacement is just 50 meters.