A thief is running away in his car with a velocity of 20m/s.A police jeep is following him which is sighted by the thief in his rearview mirror, a convex mirror of focal length 10m.He observes that the jeep is moving towards him with a speed of 1cm/s.If the magnification of the mirror for the jeep at that time is 1/10. Find:

1. actual speed of the jeep

2. rate at which magnification is changing

To tackle this problem, we need to break it down into two parts: first, we'll determine the actual speed of the police jeep, and then we'll find out how the magnification is changing over time. Let's dive into each part step by step.

Finding the Actual Speed of the Jeep

We know that the thief is moving away from the police jeep at a speed of 20 m/s, and he sees the jeep approaching him at a speed of 1 cm/s (which is 0.01 m/s). The relative speed of the jeep with respect to the thief can be expressed as:

• Relative speed = Speed of the jeep - Speed of the thief

Since the thief is moving away from the jeep, we can set up the equation:

• 1 cm/s = Speed of the jeep - 20 m/s

Converting 1 cm/s to m/s gives us 0.01 m/s. Now we can rearrange the equation to find the speed of the jeep:

• Speed of the jeep = 20 m/s + 0.01 m/s = 20.01 m/s

Calculating the Rate of Change of Magnification

Next, we need to determine how the magnification is changing. The magnification (m) for a convex mirror is given by the formula:

• m = -v/u

Where:

• v = image distance (positive for virtual images in convex mirrors)
• u = object distance (negative for real objects in convex mirrors)

From the problem, we know that the magnification is 1/10. This means:

• m = -v/u = -1/10

Let's denote the object distance (the distance of the jeep from the mirror) as u. Rearranging gives us:

• v = -u/10

Now, to find the rate of change of magnification (dm/dt), we can differentiate the magnification formula with respect to time:

• dm/dt = - (1/u) (du/dt) + (v/u^2) (dv/dt)

Since the jeep is moving towards the mirror, du/dt will be negative (as u decreases). We already know:

• du/dt = -20.01 m/s (the speed of the jeep)
• dv/dt = 0 (since the image distance does not change as the jeep approaches)

Substituting these values into the differentiated equation gives us:

• dm/dt = - (1/u)(-20.01) + (v/u^2)(0)

Now, we need to find the value of u. Using the magnification formula:

• m = -v/u = -1/10

We can express u in terms of v:

• u = -10v

Substituting v = -u/10 into the equation for u, we can find the actual distance. However, since we are interested in the rate of change of magnification, we can simplify our calculations by focusing on the relationship between the speeds and the magnification:

Thus, the rate at which magnification is changing can be calculated as:

• dm/dt = 20.01/u

To find the exact value of dm/dt, we would need the actual distance of the jeep from the mirror at that moment. However, we can conclude that the rate of change of magnification is directly proportional to the speed of the jeep and inversely proportional to the distance from the mirror.

Summary of Results

1. The actual speed of the police jeep is 20.01 m/s.

2. The rate at which magnification is changing depends on the distance of the jeep from the mirror, but it is influenced by the speed of the jeep as discussed.

Understanding these concepts helps in visualizing how relative motion and optics interact in real-world scenarios, especially in situations involving reflections and speeds.