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Motion question

2023 · Shift 1 · Q46
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Motion question

2023 · Shift 1 · Q46

JEE AdvancedPhysicsMotionNumerical+4 / −1
A person of height 1.6 m1.6 \mathrm{~m}1.6 m is walking away from a lamp post of height 4 m4 \mathrm{~m}4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is 60 cm s−160 \mathrm{~cm} \mathrm{~s}^{-1}60 cm s−1, the speed of the tip of the person's shadow on the ground with respect to the person is ‾cm s−1\underline{\hspace{2cm}}\mathrm{cm}~ \mathrm{s}^{-1}​cm s−1.
Numerical answer
View written solutionFree

Correct answer: 40

  1. Set up the geometry

Let:

  • xxx = distance of the person from the lamp post
  • yyy = length of the person's shadow

Then the distance of the tip of the shadow from the lamp post is x+yx+yx+y.

Given:

  • height of lamp post =4 m= 4\,\text{m}=4m
  • height of person =1.6 m= 1.6\,\text{m}=1.6m
  • speed of person =dxdt=60 cm s−1= \dfrac{dx}{dt} = 60\,\text{cm s}^{-1}=dtdx​=60cm s−1
  1. Use similar triangles

The large triangle has:

  • height 444
  • base x+yx+yx+y

The small triangle has:

  • height 1.61.61.6
  • base yyy

So,

4x+y=1.6y\frac{4}{x+y} = \frac{1.6}{y}x+y4​=y1.6​

Cross-multiplying,

4y=1.6(x+y)4y = 1.6(x+y)4y=1.6(x+y) 4y=1.6x+1.6y4y = 1.6x + 1.6y4y=1.6x+1.6y 2.4y=1.6x2.4y = 1.6x2.4y=1.6x y=1.62.4x=23xy = \frac{1.6}{2.4}x = \frac{2}{3}xy=2.41.6​x=32​x
  1. Differentiate with respect to time
dydt=23dxdt\frac{dy}{dt} = \frac{2}{3}\frac{dx}{dt}dtdy​=32​dtdx​

Substitute dxdt=60\dfrac{dx}{dt} = 60dtdx​=60 cm/s:

dydt=23×60=40 cm s−1\frac{dy}{dt} = \frac{2}{3}\times 60 = 40\,\text{cm s}^{-1}dtdy​=32​×60=40cm s−1
  1. Interpret the required quantity

The question asks for the speed of the tip of the shadow on the ground with respect to the person.

This is exactly the rate at which the shadow length changes, i.e. dydt\dfrac{dy}{dt}dtdy​.

Hence,

40 cm s−1\boxed{40\,\text{cm s}^{-1}}40cm s−1​
  1. Comparison with stored answer

Stored correct answer = 404040

Our derived answer also = 404040. Therefore, the answer agrees.

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