Question

If the potential energy of a particle of mass \(0.1 \, \text{kg}\) moving along the x-axis is \(5x(x - 4)\), then the speed of the particle is maximum at a position of:

• A. \( x = 2 \, \text{m} \)
• B. \( x = 3 \, \text{m} \)
• C. \( x = 0.5 \, \text{m} \)
• D. \( x = 5 \, \text{m} \)

Hint:

For a particle moving under a potential, the speed is maximum when the force acting on the particle is zero. This can be found by setting the derivative of the potential to zero.

Answer 1

The Correct Option is B

The potential energy \(U(x)\) is given as: \[ U(x) = 5x(x - 4). \] The force \(F(x)\) is the negative gradient of the potential energy: \[ F(x) = - \frac{dU}{dx} = - \frac{d}{dx} \left( 5x(x - 4) \right). \] Calculating the derivative: \[ F(x) = -5 \left( 2x - 4 \right) = -10(x - 2). \] The speed of the particle is maximum where the force is zero, which occurs at \(x = 3 \, \text{m}\).