Suppose while sitting in a parked car, you notice a jogger approaching towards you in the side view mirror of R = 2 m. If the jogger is running at a speed of 5 m s–1, how fast the image of the jogger appear to move when the jogger is (a) 39 m, (b) 29 m, (c) 19 m,and (d) 9 m away.

To determine how fast the image of the jogger appears to move in the side view mirror, we can use the concept of relative speed and the geometry of the situation. When the jogger approaches the parked car, their speed relative to the mirror will change as they get closer. Let's break this down step by step.

Understanding the Setup

The side view mirror creates an image of objects behind the car. The distance from the mirror to the jogger and the distance from the mirror to the car itself influence how quickly the image of the jogger seems to move. We’ll use the following formula to calculate the apparent speed of the jogger’s image:

Apparent Speed Formula

The apparent speed of the image (V_image) can be calculated using the formula:

• V_image = V_jogger * (D / (D - R))

Where:

• V_jogger = speed of the jogger (5 m/s)
• D = distance of the jogger from the mirror
• R = distance from the mirror to the car (2 m)

Calculating the Apparent Speed

Now, let’s calculate the apparent speed for each distance of the jogger from the mirror.

When the Jogger is 39 m Away

Using the formula:

V_image = 5 m/s * (39 m / (39 m - 2 m))

V_image = 5 m/s * (39 / 37) ≈ 5.27 m/s

When the Jogger is 29 m Away

Substituting in the new distance:

V_image = 5 m/s * (29 m / (29 m - 2 m))

V_image = 5 m/s * (29 / 27) ≈ 5.37 m/s

When the Jogger is 19 m Away

Again, we plug in the new distance:

V_image = 5 m/s * (19 m / (19 m - 2 m))

V_image = 5 m/s * (19 / 17) ≈ 5.59 m/s

When the Jogger is 9 m Away

Finally, we calculate for the closest distance:

V_image = 5 m/s * (9 m / (9 m - 2 m))

V_image = 5 m/s * (9 / 7) ≈ 6.43 m/s

Summary of Results

Here’s a quick overview of the apparent speeds at each distance:

• At 39 m: ~5.27 m/s
• At 29 m: ~5.37 m/s
• At 19 m: ~5.59 m/s
• At 9 m: ~6.43 m/s

As the jogger gets closer to the car, the speed of their image in the mirror appears to increase. This is because the angle at which the jogger is viewed changes, making it seem like they are moving faster relative to the fixed point of the car. This is a fascinating example of how perspective influences our perception of speed!