How does the missing + sign in the equation affect the trajectory of a particle?

[Hypervalent_Iodine]

Homework Statement

Consider an atom of mass m moving along the x
direction with an initial position x1 and initial speed v1. If the
atom moves for a time interval Δt in a region where the potential energy varies as V(x), what is its speed v2 at position x2?

Relevant Equations

Ek=(mv^2)/2
Fx=-dV/dx
E=Ek+V

Can someone explain how can i get that answer?

 

[PeroK]

At first sight, the equation looks like there is a + sign missing.

 

[RPinPA]

What they appear to have done (though as @PeroK points out, the answer appears to be incorrect), is calculate the acceleration and then use the equations of accelerated motion. The answer also appears to be assuming that $\Delta t$ is small, so the force and acceleration are treated as constant.

What is the relationship between potential V(x) and force? How is force related to acceleration and velocity?

 

[haruspex]

At first sight, the equation looks like there is a + sign missing.

Even that doesn’t fix it.
They have created a problem for themselves by referring to speed instead of velocity.
In velocities it would be $\vec v_2=\vec v_1-\frac{dV}{dx}\frac{\Delta t}m\hat x$.
To rewrite that in terms of speeds ($|\vec v_i|$) would involve breaking it into cases.