how has T=0.5mv^2 derived from W=Fs

The equation T=0.5mv^2 is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Here's the step-by-step derivation:

1. Work (W) is defined as the force (F) applied on an object times the distance (s) over which it is applied. So, W = F * s.

2. Force (F) is also defined as mass (m) times acceleration (a). So, F = m * a.

3. Acceleration (a) is the change in velocity (v) over time, or in this case, it's the final velocity (v) assuming the object started from rest. So, a = v / t.

4. Substituting F = m * a and a = v / t into the work equation gives W = m * v * s / t.

5. The distance (s) traveled by an object under constant acceleration starting from rest is given by s = 0.5 * v * t.

6. Substituting s = 0.5 * v * t into the work equation gives W = m * v * 0.5 * v * t / t.

7. The t in the numerator and denominator cancel out, leaving W = 0.5 * m * v^2.

8. The work done on an object is equal to the change in its kinetic energy, so the kinetic energy (T) of the object is T = 0.5 * m * v^2.

So, the equation T=0.5mv^2 is derived from the work-energy theorem, not directly from W=Fs.