An object starts at \(x=x_{0}\) with a velocity \(v=v_{0}\) at the time \(t=t_{0}\) and moves with a constant acceleration \(a_{0}\). Show that the velocity \(v\) when the object has moved to a position \(x\) is \(v^{2}-v_{0}^{2}=2 a_{0}\left(x-x_{0}\right)\).

In physics, constant acceleration means an object's velocity changes at a consistent rate over time. When an object experiences constant acceleration, its speed increases by the same amount each second.
To describe this motion, we use kinematic equations that relate position, velocity, acceleration, and time. One such equation is:
\[ v = v_0 + a_0 t \]
Here:

• \(v\) is the final velocity
• \(v_0\) is the initial velocity
• \(a_0\) is the constant acceleration
• \(t\) is the time elapsed

Constant acceleration allows us to predict the future position and velocity of an object. By using these equations, we can solve for unknown variables if the other quantities are known.
For example, if you know the initial and final velocities, and the distance traveled, you can find the acceleration using algebraic manipulation.

The work-energy theorem helps us understand the relationship between the work done on an object and its kinetic energy. The theorem states:
\[ \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 = F d \]
Here:

• \(m\) is the mass of the object
• \(v\) is the final velocity
• \(v_0\) is the initial velocity
• \(F\) is the force applied
• \(d\) is the distance over which the force is applied

When an object moves under a force, its kinetic energy changes. The work done by the force is equal to this change in kinetic energy.
Let's connect this to our earlier problem. For an object with a constant acceleration \(a_0\), the force \(F\) is given by \(m a_0\) and the distance \(d\) is \(x - x_0\). Substitute these values into the work-energy theorem:
\[ \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 = m a_0 (x - x_0) \]
By cancelling out the mass \(m\), we get back to:
\[ \frac{1}{2} v^2 - \frac{1}{2} v_0^2 = a_0 (x - x_0) \]
and multiplying through by 2 results in:
\[ v^2 - v_0^2 = 2 a_0 (x - x_0) \]

Algebraic manipulation is a crucial skill in physics, allowing us to rearrange and solve equations to find unknowns. Let's break down how we use algebra in our solution:
First, we used the kinematic equation for position:
\[ x = x_0 + v_0 t + \frac{1}{2} a_0 t^2 \]
We isolated \(t\) by rearranging terms:
\[ x - x_0 = v_0 t + \frac{1}{2} a_0 t^2 \]
While solving for \(t\) directly in this equation is complex, we used another method by applying the work-energy theorem.
Through algebraic steps, we substituted known values, and simplified:
\[ \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 = m a_0 (x - x_0) \]
We cancelled out the mass \(m\), making the equation simpler:
\[ \frac{1}{2} v^2 - \frac{1}{2} v_0^2 = a_0 (x - x_0) \]
Multiplying through by 2, we finally isolated the variable \(v^2\):
\[ v^2 - v_0^2 = 2 a_0 (x - x_0) \]
Algebraic manipulation thus helps in simplifying and solving complex physics problems by isolating and focusing on necessary variables.