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Question

A car is fitted with a convex side-view mirror of focal length 20cm. A second car 2.8m behind the first car is overtaking the first car at a relative speed of 15m/s. The speed of the image of the second car as seen in the mirror of the first one is:


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Solution

Step 1: Given

  1. Focal length of the convex mirror is 20cm which is fitted on the first car

  1. Second car 2.8m behind the first car
  2. Second car overtake the fast car at a relative speed of 15m/s

Step 2: Formula used

  1. 1v+1u=1f where u is the distance of the object from the mirror, v is the distance of the image from the mirror and f is the focal length

Step 3: Find the the distance of the second car

Here, we are aware that the focal length is f=+20cm. Our object distance, denoted by u, where u=-280cm, is 2.8m, or behind the first car. The first vehicle is moving at a 15m/s relative speed.

So dudt=15m/s

Now, we utilize the mirror formula to determine the image distance; specifically, the second car's distance is indicated by the symbol v, and v may be calculated as follows:

1v+1u=1f1v=1f-1u1v=u-fuf1v=-280-20-280·20v=28015cm

Here, u is the distance of the first car, and f is the focal length of the second car’s side view mirror.

Step 4: Find the speed of the image of the second car as seen in the mirror of the first one

We have

dudt=15m/s

v=28015cmu=280cm

The second car's speed will now be determined by differentiating the mirror formula with regard to time in the manner shown below:

ddt1u+ddt1v=ddt1f

But the focal length of the mirror remains constant; hence its derivative will be zero.

dudt-1u2+dvdt-1v2=0dudt-1u2=dvdt1v2dvdt=-v2u2dudtdvdt=-280152280215dvdt=-115m/s

Hence the required answer is -115m/s


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