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Concepts Of Physics MCQ Edition [Volume 1]PhysicsWave Motion and Waves on a String

A wave is described by the equation y = (1.0 mm ) ( x 2.0 cm - t 0.01 s ) . What are the speeds of the particles at x = 3.0 cm , 5.0 cm and 7.0 cm at t = 0.01 s ?

Options

  1. A10 cm s ⁻¹ for all three positions
  2. Bzero for all three positions
  3. C5 cm s ⁻¹ for all three positions
  4. Dzero, 5 cm s ⁻¹ , and 10 cm s ⁻¹ respectively

Correct answer

B. zero for all three positions

Step-by-step solution

The given wave equation is y = (1.0 mm ) ( x 2.0 cm - t 0.01 s ) . The particle velocity v_p is given by the partial derivative of y with respect to t : v_p = y t = (1.0 mm ) ( x 2.0 - t 0.01 ) (- 0.01 ) Since 1.0 mm = 0.1 cm , we get: v_p = -10 ( x 2.0 - t 0.01 ) cm s ⁻¹ At t = 0.01 s , the term t 0.01 = 1 . Substituting this into the velocity equation: v_p = -10 ( x 2.0 - ) = 10 ( x 2.0 ) cm s ⁻¹ Evaluating the particle speed at the given positions: For x = 3.0 cm , v_p = 10 ( 3 2 ) = 0 For x = 5.0 cm , v_p = 10

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