Question

A particle is moving on x-axis has potential energy $U = 2-20X + 5x^2$ Joules along x-axis. The is released at x = -3. The maximum value of 'x' will be [x is in meters and U is in joules]

• A. 5 m
• B. 3 m
• C. 7 m
• D. 8 m

Answer 1

Answer: (C)

7 m

Answer 2

The problem asks for the maximum value of 'x' reached by a particle moving under a given potential energy function, starting from a specific point with zero initial kinetic energy. We can solve this using the principle of conservation of mechanical energy.

1. Calculate the total mechanical energy (E) of the particle:

The particle is released at $x = -3$. This implies that its initial kinetic energy ($K_{initial}$) at this point is zero. The potential energy function is given by $U = 2 - 20x + 5x^2$ Joules. First, calculate the potential energy at the release point $x = -3$: $U(-3) = 2 - 20(-3) + 5(-3)^2$ $U(-3) = 2 + 60 + 5(9)$ $U(-3) = 2 + 60 + 45$ $U(-3) = 107$ Joules

The total mechanical energy ($E$) is the sum of potential and kinetic energy. Since the kinetic energy at release is zero: $E = U_{initial} + K_{initial}$ $E = 107 + 0 = 107$ Joules

2. Determine the turning points:

As the particle moves, its total mechanical energy remains constant (assuming no non-conservative forces like friction). The particle will oscillate between two extreme points where its kinetic energy becomes zero. These are called turning points. At these points, all the energy is potential energy. So, at the turning points, $U(x) = E$. $2 - 20x + 5x^2 = 107$

Rearrange the equation into a standard quadratic form: $5x^2 - 20x + 2 - 107 = 0$ $5x^2 - 20x - 105 = 0$

Divide the entire equation by 5 to simplify: $x^2 - 4x - 21 = 0$

3. Solve the quadratic equation for x:

We can solve this quadratic equation by factoring. We need two numbers that multiply to -21 and add up to -4. These numbers are -7 and 3. So, the equation can be factored as: $(x - 7)(x + 3) = 0$

This gives two possible values for x: $x - 7 = 0 \Rightarrow x = 7$ meters $x + 3 = 0 \Rightarrow x = -3$ meters

4. Identify the maximum value of x:

The two turning points are $x = -3$ m and $x = 7$ m. The particle was released at $x = -3$ m, which is one of the turning points. The other turning point, $x = 7$ m, represents the maximum value of 'x' reached by the particle.