Problem 41 - University Physics with Modern Physics

Question

You are driving down a straight highway at a speed of \(v=50.0 \mathrm{~m} / \mathrm{s}\) relative to the ground. An oncoming car travels with the same speed in the opposite direction. With what relative speed do you observe the oncoming car?

Step-by-Step Solution

Answer

Answer: The relative speed between the two cars is 100.0 m/s.

Step 1: Given values

The given values are: - Speed of car 1 relative to the ground, \(v_1 = 50.0 \mathrm{~m/s}\) - Speed of car 2 relative to the ground, \(v_2 = 50.0 \mathrm{~m/s}\)

Step 2: Calculate relative speed between the two cars

Since both cars are moving in opposite directions, we simply add their individual speeds relative to the ground to determine the combined relative speed between them. $$v_\text{relative} = v_1 + v_2$$ Plugging in the given values, we get: $$v_\text{relative} = 50.0 \mathrm{~m/s} + 50.0 \mathrm{~m/s}$$

Step 3: Solve for relative speed

Adding the speeds together, we find the relative speed between the two cars: $$v_\text{relative} = 100.0 \mathrm{~m/s}$$ This means the oncoming car is observed to be traveling at a relative speed of \(100.0 \mathrm{~m/s}\) with respect to the other car.

Key Concepts

Understanding Kinematics

Kinematics is a branch of physics that deals with motion. It helps us understand how objects move, describing their velocity, acceleration, and displacement without considering the forces that cause them to move. In this problem, kinematics allows us to analyze the motion of two cars on a highway.

• Velocity describes how fast an object moves and in what direction.
• Speed is the magnitude of velocity and ignores direction.

By focusing on speed and direction, kinematics makes it easier for us to understand how objects behave when in motion, such as our cars traveling in opposite directions.

Exploring Relative Motion

Relative motion refers to how the motion of one object is perceived from another's point of view. It highlights that motion isn't absolute; instead, it's always related to something else. Imagine standing on the highway while two cars zoom past one another. Each driver sees the other car moving faster than their own speed relative to the ground.

• When two objects move towards each other, their speeds are added.
• When moving in the same direction, their relative speed is the difference between their speeds.

In our example, each car sees the other approaching at a relative speed much greater than their individual ground speed. This concept of relative motion helps us describe how fast objects move in relation to one another.

The Principle of Velocity Addition

The principle of velocity addition allows us to determine the combined speed of two moving objects as seen from one another. Using this, we can simplify calculations involving relative speeds.

In our exercise, each car's speed is 50.0 m/s relative to the ground. As they are moving in opposite directions, we apply the velocity addition principle to compute their relative velocity:
\[v_\text{relative} = v_1 + v_2 = 50.0 \, \text{m/s} + 50.0 \, \text{m/s} = 100.0 \, \text{m/s}\]

• The cars' speeds are summed because they move towards each other.
• This method provides a straightforward way to understand how quickly they approach one another.

Velocity addition simplifies problems by offering a clear way to determine combined speeds, making it easier to understand interactions in relative terms.