The kinetic energy ($E_k$) of a particle moving along the X-axis varies with its position ($X$) as shown in the figure. The force acting on the particle at $X = 10$ m is
The force acting on a particle can be derived from the potential energy function, as the force is the negative gradient of the potential energy. In terms of kinetic energy, the total mechanical energy \(E\) is conserved, so: \[ E = E_k + U(x), \] where \(E_k\) is the kinetic energy and \(U(x)\) is the potential energy. In this case, the force \(F\) can be obtained from the relation: \[ F = - \frac{dE_k}{dx}. \] From the graph, we can observe the variation of kinetic energy \(E_k\) with respect to position \(x\). To calculate the force at \(x = 10 \, \text{m}\), we find the slope of the curve at that point. At \(x = 10 \, \text{m}\), from the graph, the slope of the kinetic energy curve is negative and equals \(-5 \, \text{N}\). Thus, the force acting on the particle at \(x = 10 \, \text{m}\) is: \[ \boxed{-5i \, \text{N}}. \]
