Simple harmonic oscillations: (e) velocity v = ±ω√(A² - x²) hence vmax = ωA

The idea

For a simple harmonic oscillator the speed at displacement x is given by v = ±ω√(A² - x²), where A is the amplitude and ω is the angular frequency. The ± sign shows that the oscillator passes through a given displacement in either direction. Speed is greatest at the equilibrium position, where x = 0, giving vmax = ωA. Speed is zero at the extremes of the motion, where x = ±A, because the oscillator momentarily stops before reversing. The equation follows from energy conservation or from combining the sinusoidal displacement and velocity expressions, and it allows the speed to be found at any displacement without knowing the time.

What earns the marks

• States or applies v = ±ω√(A² - x²) with A as amplitude, x as displacement and ω as angular frequency.
• Recognises that vmax = ωA occurs at x = 0, the equilibrium position.
• Recognises that v = 0 at x = ±A, the extremes of the oscillation.
• Interprets the ± sign as showing the two possible directions of motion at a given displacement.
• Calculates ω from the period or frequency before substituting into the velocity equation.

Common mistakes and how to put them right

• Using vmax = ωA² or vmax = A/ω.

  Put it right: Use vmax = ωA, which has units of m s⁻¹.

• Substituting x = A into the velocity equation and expecting a maximum.

  Put it right: Note that v = 0 at x = ±A.

• Treating the ± sign as an error.

  Put it right: Explain that it represents motion in either direction through the same displacement.

• Using frequency f in place of angular frequency ω.

  Put it right: Convert with ω = 2πf.

Exam tips

• Check that ω has units of rad s⁻¹ before substituting into the velocity equation.
• Use vmax = ωA as a quick check on the maximum speed at equilibrium.
• Keep the square root until the final step to avoid premature rounding.