Question

You ride your bike along a straight line from your house to a store \(1000 .\) m away. On your way back, you stop at a friend's house which is halfway between your house and the store. a) What is your displacement? b) What is the total distance traveled? After talking to your friend, you continue to your house. When you arrive back at your house, c) What is your displacement? d) What is the distance you have traveled?

Short answer

Answer: The displacement after the entire trip is 0m, and the total distance traveled is 2000m.

Step-by-step solution

Part (a): Find displacement from house to store

For displacement, we consider the starting point and the final point. Since the straight-line distance from the house to the store is 1000m, the displacement will be equal to this distance: 1000m.

Part (b): Calculate the total distance traveled from house to the store and then to the friend's house

The distance traveled is the total path length. As the bike ride is along a straight line from the house to the store, and then returning to a point halfway between them, the distance traveled from the house to the store is 1000m, and the distance from the store to the friend's house is half of this distance, 1000m / 2 = 500m. Thus, the total distance traveled so far is 1000m + 500m = 1500m.

Part (c): Determine the displacement after talking to the friend and continuing to the house

The displacement, being the change in position, is the difference between the final position and the starting position. The starting position is the house, and the final position is the house again. Therefore, the displacement after the entire trip is 0m.

Part (d): Find the total distance traveled after talking to the friend and continuing to the house

The distance traveled is the total path length. The distance from the friend's house (halfway point) to the house is the same as the distance from the friend's house to the store, which is 500m. Therefore, the total distance traveled during the entire trip is the distance traveled in part (b) plus the distance from the friend's house to the starting house, which amounts to 1500m + 500m = 2000m.

Key concepts

Vector and Scalar Quantities

Understanding the difference between vector and scalar quantities is fundamental in physics, especially when studying motion. Scalar quantities are described by magnitude alone. Common scalar quantities include distance, mass, speed, energy, and time. They tell us how much there is of something, but not in which direction it occurs.

On the other hand, vector quantities have both magnitude and direction. Examples of vector quantities include displacement, velocity, acceleration, and force. They are represented by arrows where the length of the arrow shows the magnitude and the arrow points in the direction of the quantity. For example, if our exercise states that you rode your bike 1000 meters to the east, the displacement would be a vector quantity showing both the distance (1000 meters) and the direction (east).

Understanding that displacement is a vector quantity while distance is a scalar quantity helps explain why the displacement can be zero after a round trip (since you end up where you started), while the distance traveled is the total path length covered, regardless of the direction.

Kinematics in One Dimension

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. When we study kinematics in one dimension, we're looking at motion along a straight line. This simplifies problems to a single axis of motion - often labeled the x-axis for horizontal motion or the y-axis for vertical motion.

In the case of the exercise, your bike ride can be considered one-dimensional motion, back and forth along the same straight path from your house to the store. One-dimensional kinematics involves calculating various parameters, such as displacement, velocity, and acceleration, along this straight line. Even though your total distance is the sum of all movements along that line, your displacement only considers your initial and final positions relative to the chosen axis.

Equations of motion in one dimension are vital tools to predict the future position of an object, how long it will take to get there, and its velocity at any point in time. This assumes a constant acceleration, which isn't provided in the exercise, but when working on more complex one-dimensional problems, these equations become essential.

Distance versus Displacement

The concept of distance versus displacement often confuses students because they both relate to 'how far' an object has moved, but they are not the same.

Distance is a scalar quantity that refers to how much ground an object has covered during its motion. It's the total path length traveled regardless of the direction and can never be negative. In the exercise, when you ride your bike to the store and back to your friend's house, and eventually return home, each segment adds to the total distance traveled, resulting in 2000m.

Displacement, however, is the vector quantity that refers to how far out of place an object is; it's the object's overall change in position. Displacement considers the direction of the movement. In the exercise, despite traveling a distance of 2000m, your displacement is 0m when you return to your house because the starting and ending points are the same, showing that displacement is direction-sensitive.

Clarifying the concepts of distance and displacement is crucial when analyzing motion since they affect how we apply the kinematic equations and understand an object's movement through space.