Problem 68 - College Physics

Question

While you are traveling in a car on a straight, level interstate highway at \(90 \mathrm{~km} / \mathrm{h}\), another car passes you in the same direction; its speedometer reads \(120 \mathrm{~km} / \mathrm{h}\). (a) What is your velocity relative to the other driver? (b) What is the other car's velocity relative to you?

Step-by-Step Solution

Answer

Your velocity relative to the other driver is -30 km/h. The other car's velocity relative to you is 30 km/h.

Step 1: Identify the Given Velocities

The velocity of your car is given as \( 90 \text{ km/h} \). The velocity of the other car is \( 120 \text{ km/h} \). Both velocities are in the same direction on a straight, level interstate highway.

Step 2: Determine Relative Velocity (a)

To find your velocity relative to the other driver, subtract the velocity of your car from the velocity of the other car. This is done using the formula: \[ v_{relative} = v_{other} - v_{your} = 120 \text{ km/h} - 90 \text{ km/h} \].

Step 3: Calculate Relative Velocity (a)

Using the formula from step 2, calculate the result: \[ v_{relative} = 120 \text{ km/h} - 90 \text{ km/h} = 30 \text{ km/h} \]. This means your velocity relative to the other driver is \(-30\text{ km/h}\), indicating you are moving slower.

Step 4: Determine Relative Velocity (b)

To find the other car's velocity relative to you, subtract your velocity from the other car's velocity, similar to step 2 but inverted: \[ v_{relative} = v_{your} - v_{other} = 90 \text{ km/h} - 120 \text{ km/h} \].

Step 5: Calculate Relative Velocity (b)

Using the formula from step 4, calculate the result: \[ v_{relative} = 90 \text{ km/h} - 120 \text{ km/h} = -30 \text{ km/h} \]. This implies the other car's velocity relative to you is \(30 \text{ km/h}\), indicating it is moving faster.

Key Concepts

Velocity Calculation

Understanding how to calculate velocity is crucial in physics, as it helps us comprehend motion. Velocity is a vector, meaning it has both magnitude and direction. In this exercise, when determining the relative velocity between two cars, we need to look at how their velocities are compared to each other.

To compute the relative velocity of one object in relation to another, you simply subtract one velocity vector from the other.

• For instance, to determine your velocity relative to another driver whose car moves faster, use the formula: \[ v_{relative} = v_{other} - v_{your} \].
• This results in a positive value if you are slower, as shown in \(30 ext{ km/h}\). This signifies that the other car is progressing ahead of you.

Conversely, calculating from your car’s perspective to the other driver, the formula \[ v_{relative} = v_{your} - v_{other} \] results in a negative value. This time, the magnitude shows how much faster the other vehicle travels in comparison to you.

These calculations are straightforward once you understand the basic principle of subtracting velocities, taking into account their direction.

Frame of Reference

The concept of a frame of reference is fundamental in physics to understand motion. It essentially means the point of view from which you are observing the motion. Different frames of reference can dramatically change the perceived velocity of an object.

In our exercise, each car represents a different frame of reference. When viewing from your car, which is considered a moving reference point,

• You see the other car overtaking you with a relative velocity of \(30 ext{ km/h}\).
• The other vehicle appears to be moving away faster because you're stationary compared to it.

However, in the other driver's frame of reference:

• You appear to be moving backwards at \(30 ext{ km/h}\), because you are slower. They perceive you moving in reverse relative to their car.

Understanding frames of reference helps you comprehend how motion is observed and described from different perspectives. It highlights that velocity isn't absolute but varies according to who or what is observing it.

One-Dimensional Motion

The exercise involves one-dimensional motion because both cars are moving in a straight line along the highway. One-dimensional motion means that the movement occurs in a single axis.

In one-dimensional motion, the complexity of the motion is reduced as it involves:

• Linear paths – Straight lines with no curves.
• Easy calculations – no need to consider changes in direction or multiple axes.

Calculating velocities becomes simpler in this context as you only have one directional component to account for.

When analyzing our scenario:

• Even though your car is at \(90 ext{ km/h}\), and the other at \(120 ext{ km/h}\), you both move on the same straight path. This makes it easy to find relative velocities, because there's no need to consider multiple dimensions.

Understanding one-dimensional motion is vital because it forms the foundation for more complex motion analysis, involving additional dimensions and eventual curves in paths.