Problem 77 - University Physics with Modern Physics

Question

cp calc You are in your car driving on a highway at 25 \(\mathrm{m} / \mathrm{s}\) when you glance in the passenger-side mirror (a convex mirror with radius of curvature 150 \(\mathrm{cm}\) ) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of 1.9 \(\mathrm{m} / \mathrm{s}\) when the truck is 2.0 \(\mathrm{m}\) from the mirror, what is the speed of the truck relative to the highway?

Step-by-Step Solution

Answer

The truck's speed relative to the highway is approximately 31.1 m/s.

Step 1: Understand the given information

We know that the car is moving at 25 \( \mathrm{m/s} \). The radius of curvature of the convex mirror is 150 \( \mathrm{cm} \) (which is 1.5 \( \mathrm{m} \)). The truck is moving towards the mirror at 1.9 \( \mathrm{m/s} \) when it is 2.0 \( \mathrm{m} \) from the mirror.

Step 2: Relate image speed to object speed

For a convex mirror, the relation between the speed of the image (\( v' \)) and the speed of the object (\( v \)) is given by the mirror formula \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the mirror. We use the derivative characterizing speed changes: \( \frac{d_i}{d_o} \times v = v' \).

Step 3: Calculate the focal length

The focal length \( f \) of a convex mirror is half the radius of curvature, but with a negative sign (since it's a convex mirror): \( f = -\frac{R}{2} = -0.75 \) \( \mathrm{m} \).

Step 4: Mirror Formula Application

At the given instance, with \( d_o = 2.0 \ \mathrm{m} \) and using the mirror formula \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), solve for \( d_i \) to use \( \frac{1}{-0.75} = \frac{1}{2.0} + \frac{1}{d_i} \), solve it to find \( d_i \).

Step 5: Calculate speed

Use the relation \( \frac{v}{v'} = \left(\frac{d_o}{d_i}\right)^2 \) to find the object speed \( v \) in terms of \( v' \) which is 1.9 \( \mathrm{m/s} \), plug \( v' \) and the found \( d_o \) and \( d_i \) to calculate \( v \).

Step 6: Find truck speed relative to highway

The relative speed of the truck to the car is \( v_{relative} = v + 25 \), then find \( v_{highway} \) using calculated \( v \) from previous steps, yielding the speed on highway as \( v_{highway} = v_{relative} + 25 \).

Key Concepts

Mirror Formula

When dealing with mirrors, particularly convex mirrors like the one in this problem, we use the "mirror formula" to establish a relationship between the object distance, image distance, and the focal length of the mirror. The formula is expressed as:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

The variable \( f \) represents the focal length of the mirror, \( d_o \) is the distance between the object and the mirror, and \( d_i \) indicates the image distance from the mirror. In the context of convex mirrors, the focal length is negative because it diverges light rays rather than converging them. This formula allows us to calculate unknown variables when others are provided, assisting in understanding the relationship between objects and their reflections in curved surfaces. For our problem, we know \( d_o \) and can solve for \( d_i \).

Image Object Speed Relation

In this type of problem, we often relate the speed of the observed image to the speed of the actual object using a derived formula linked to the mirror formula. The derivation is given by:

• \( \frac{d_i}{d_o} \times v = v' \)

Where \( v \) is the speed of the truck (object), \( v' \) is the speed of the image as it appears in the mirror, \( d_o \) is the distance of the truck from the mirror, and \( d_i \) is the distance of the image from the mirror. Solving this equation requires knowledge of the image's apparent speed and the mirror's characteristics, such as its focal length. This understanding is crucial for calculations involving motion relative to a reflective surface.

Focal Length Calculation

For a convex mirror, the focal length \( f \) can be calculated easily as it is half the radius of curvature, but always negative. The formula for determining the focal length is:

• \( f = -\frac{R}{2} \)

Where \( R \) is the radius of curvature. In this problem, the radius of curvature is given as 1.5 meters, leading to a focal length of \( f = -0.75 \) meters. By inserting this value into the mirror formula, we can determine the image distance \( d_i \) which then helps us prepare for subsequent calculations involving speed and distance relationships.

Relative Speed Calculation

The relative speed calculation in our problem involves determining the real speed of the truck relative to different reference points, like the mirror and the highway. After calculating the speed of the truck using the image-object speed relation and the gathered image distance, we then compile this with the car's speed.
First, calculate the truck's speed relative to the car, \( v_{relative} = v + 25 \), where 25 m/s is the speed of the car. After that, to find the truck's speed relative to the highway, add this figure to the car's speed for a complete understanding of the truck's movement on the road system: \( v_{highway} = v_{relative} + 25 \). This step helps clarify how the truck is truly moving in space, which is invaluable for real-world applications and problem solving in physics involving motion dynamics.