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Chapter 4: Problem 88

A white automobile is traveling at a constant specd of \(90 \mathrm{~km} / \mathrm{h}\) on a highway. The driver notices a red automobile. 1.0 km behind, traveling in the same direction. 'Two minutes later, the red automobile pasoes the white automobile. (a) What is the average speed of the red automobile relative, to the white? (b) What is the speed of the red automobile relative to the ground?

Short Answer

Expert verified
(a) 30 km/h, (b) 120 km/h.

Step by step solution

01

Convert Time to Hours

The time given is two minutes, which needs to be converted into hours for consistency with speed units (km/h). Since there are 60 minutes in an hour, 2 minutes is equal to \(\frac{2}{60} = \frac{1}{30}\) hours.
02

Determine Relative Speed

In two minutes, the red car covers the 1 km gap and manages to overtake the white car, which leads to the determination of the relative speed. The relative speed is the additional speed the red car has over the white car: \(s_{relative} = \frac{1 \, \text{km}}{\frac{1}{30} \, \text{h}} = 30 \, \text{km/h}\).
03

Calculate Absolute Speed of the Red Car

Since the white car is moving at 90 km/h, the speed of the red car relative to the ground can be calculated by adding the relative speed to the speed of the white car: \(s_{red} = s_{white} + s_{relative} = 90 \, \text{km/h} + 30 \, \text{km/h} = 120 \, \text{km/h}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Speed
When we talk about average speed, we are referring to the total distance covered over a given period of time. It's a useful measure, especially when speed doesn't stay consistent. For this exercise, it's important to recognize that the average speed of the red automobile isn't just the speed calculated at the end of the chase, but a representation of its speed relative to the white car over the two minutes it took to catch up.
  • Total distance here is 1 km, the gap closed between the red and white cars.
  • Total time was 2 minutes, converted to hours as \( \frac{1}{30} \) hours.
  • The average speed relative to the white car is computed as \( \frac{1 \, \text{km}}{\frac{1}{30} \, \text{h}} = 30 \, \text{km/h} \).
Thus, understanding average speed helps in the calculation of how one object moves relative to another over a given period.
Speed Conversion
Speed conversion is essential in physics, particularly when working with problems involving different time units. In this exercise, time must be in the same unit as speed, typically hours, because the speed is given in km/h. Here鈥檚 how the conversion helped:
  • Given: 2 minutes must be converted to hours for consistency with km/h.
  • Calculation: \( 2 \text{ minutes} = \frac{2}{60} = \frac{1}{30} \text{ hours} \).
This conversion allows us to reliably calculate speeds and distances in consistent units, ensuring that our results for relative speeds and absolute speeds (like in our exercise) are accurate and understandable.
Problem Solving in Physics
Mastering problem solving in physics involves understanding the interplay between concepts and the ability to convert those into practical calculations. The approach to solving the problem in the exercise can be broken down as:
  • Recognizing the given data: White car's speed and the distance to be overtaken.
  • Converting necessary units for consistent calculations, such as time into hours.
  • Determining the relative speed by calculating how much faster one car is compared to the other.
  • Finally, using relative speed to find absolute speed: Adding the relative speed to the known speed of the white car to find the red car's speed relative to the ground (\( 90 \, \text{km/h} + 30 \, \text{km/h} = 120 \, \text{km/h} \)).
This approach demonstrates the critical thinking needed to navigate complex physics problems, ensuring a comprehensive understanding and application of the rules and formulas.

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Most popular questions from this chapter

According to the Gwinness Book of World Recerds, during a catastrophic explosion in Halifax on December 6,1917, William Becher wae thrown throuph the air for nome \(1500 \mathrm{~m}\) and was found, itill alive, in a tree. Aseume that Becluer left the ground and rctumed to the gacund (ignore the height of the trcc) at an anple of \(45^{\circ}\). With what sipeed did he leave the pround? How high did he rise? How long did he stay in flight?

Duting a famous jump in Richmond, Virginia, in 1903 , the, horee Heatherbloom with its rider juruped over an cbstacle \(8 \mathrm{ft} 8\) in. high whilc covering a horizontal distance of \(37 \mathrm{ft}\). At what angle and with what ipeed did the horse leave the ground? Make the (somewhat doubtful) ussumption that the motion of the horsc is particle motion.

The components of the position vector of a particle moving in the \(x-y\) plane are $$ x=A \cos b f \quad y=B t $$ where \(A, b\), and \(B\) ax constants. What are the components of the instantaneous velocity vector? The instantaneous acceleration vector? What is the speed of the particle?

You are driving an automobile at a steady speed of \(90 \mathrm{~km} / \mathrm{h}\) along a straight highway. Ahead of you is a 10-m-long truck traveling at a steady specd of \(60 \mathrm{~km} / \mathrm{h}\). You decide to pass this tracks, and you switch into the passing lane when \(40 \mathrm{~m}\) behind the truck. (a) What is your speed in the reference frame of the truck? (b) How long do you take to pass the oruck, starting \(40 \mathrm{~m}\) behind the truck and coding \(40 \mathrm{~m}\) ahead of the truck? (Hint) Calculate this time in the reference frame of the truck.)

For whet launch angle will the height and garage of a projectile be equal?
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