Question

You unconsciously estimate the distance to an object from the angle it subtends in your field of view. This angle θ in radians is related to the linear height of the object h and to the distance d by θ= h/d. Assume that you are driving a car and that another car, 1.50 m high, is 27.0 m behind you. Based on its angular size, how far away does the following car appear to be?

Answer 1

Using the relationship between angle, object height, and distance (θ = h/d), the following car that is 1.50 meters high and subtends the same angle at a distance of 27 meters, appears to be 27 meters away.

Step-by-step explanation:

The question pertains to the concept of estimating distances based on the angle subtended by an object at the eye, which is a fundamental topic in physics, specifically in geometrical optics. To find how far away the following car appears to be based on its angular size, one can use the provided relationship θ = h/d, where θ is the angle in radians, h is the linear height of the object, and d is the distance. Given that the height of the car (h) is 1.50 meters, and assuming it subtends the same angle as it would 27.0 meters behind you (since no angle is provided), the apparent distance (d) would be the same as the actual distance, which is 27.0 meters.

Answer 2

Using the formula ϴ = h/d, the perceived distance to another car of height 1.50 m, based on an estimated angular size of 0.01 radians discernible by the human eye, would be approximately 150 meters.

Step-by-step explanation:

The student's question involves calculating the perceived distance of an object based on its angular size, which is a concept from Physics related to optics and geometric proportions.

In this case, the formula ϴ = h/d can be used to find the perceived distance (d) to another car. The car's height (h) is 1.50 m, and since we want to find the perceived distance (d) based on the angular size (ϴ), we can rearrange the formula to d = h/ϴ.

Assuming an average human eye can discern objects that subtend an angle of about 0.01 radians, we can perform the calculation:

d = 1.50 m / 0.01 radians

d = 150 m

Thus, the following car would appear to be approximately 150 meters away based on the angular size perceived by the human eye.