Problem 59 - College Physics

Question

You are driving eastbound on the interstate at \(70 \mathrm{mi} / \mathrm{h}\). You observe that you are approaching a truck in your lane at a relative speed of \(20 \mathrm{mi} / \mathrm{h}\). (a) How fast is the truck moving relative to the highway? (b) If the truck were instead traveling at this speed in the westbound lane, what would be the relative velocity between you and the truck?

Step-by-Step Solution

Answer

(a) 50 mi/h. (b) 120 mi/h.

Step 1: Determine the Truck's Speed Relative to the Highway

To find the truck's speed relative to the highway, subtract your observed relative speed from your speed. You are traveling at \(70 \, \text{mi/h}\), and the relative speed to the truck is \(20 \, \text{mi/h}\). \[\text{Truck's Speed} = 70 \, \text{mi/h} - 20 \, \text{mi/h} = 50 \, \text{mi/h}.\]Thus, the truck is traveling at \(50 \, \text{mi/h}\) relative to the highway.

Step 2: Calculate the Relative Speed if the Truck is Westbound

If the truck is traveling in the opposite (westbound) direction, its speed adds to your speed to determine the relative speed. You are traveling at \(70 \, \text{mi/h}\), and we've already determined the truck's speed is \(50 \, \text{mi/h}\).The relative speed is the sum of these speeds:\[\text{Relative Speed} = 70 \, \text{mi/h} + 50 \, \text{mi/h} = 120 \, \text{mi/h}.\]Thus, if the truck is in the westbound lane, the relative speed between you and the truck is \(120 \, \text{mi/h}\).

Key Concepts

Road Traffic Physics

Road traffic physics is about understanding the motion and interaction of vehicles on roads. These principles help us predict how cars move, speed up, slow down, and interact with each other. In simple terms, it deals with how vehicles behave on highways, streets, and other transport systems.

A key aspect of road traffic physics includes **relative velocity**, where we examine how different vehicles' speeds compare to each other. This is crucial for safe driving.

• Knowing how fast you're approaching another vehicle helps in making informed decisions like when to change lanes or adjust your speed.
• It's also essential for traffic regulation, ensuring that vehicles maintain safe distances and avoid collisions.

By mastering these concepts, drivers can improve their situational awareness on the road, promoting safety for everyone.

Speed Calculation

Speed calculation is a fundamental concept in road traffic physics. It involves determining how fast an object is moving within a specific time frame. Generally, speed is calculated as:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]
In the context of vehicles, speed helps us assess how quickly a car can travel from one point to another.
For example, when you are told your relative speed to another vehicle is 20 mi/h, it means your car is moving 20 miles faster every hour than the other vehicle. To find the truck's speed relative to the highway, we subtract the relative speed from your speed:

• Your speed: 70 mi/h
• Relative speed to truck: 20 mi/h
• Truck's speed: 70 mi/h - 20 mi/h = 50 mi/h

This simple arithmetic helps us ensure we're not misjudging how fast the vehicles are traveling on the highway.

Directional Velocity

Directional velocity not only considers how fast an object is moving but also the direction in which it's heading. In road traffic scenarios, this concept is vital for understanding and predicting collisions and interactions between vehicles. When vehicles travel in opposite directions, their velocities can combine, leading to a higher relative velocity.

For instance, if you're driving eastbound at 70 mi/h, and a truck is westbound at 50 mi/h, the relative velocity isn't just the truck's speed, but the combination of both speeds:

• Your speed: 70 mi/h
• Truck's speed: 50 mi/h
• Relative velocity (opposite directions): 70 mi/h + 50 mi/h = 120 mi/h

Such calculations help drivers anticipate and react to potential hazards more effectively. Understanding directional velocity allows you to process complex driving environments with multiple moving parts and head a step towards safer driving habits.