Understanding the Displacement Equation
The given equation for the displacement of a particle moving along the x-axis is:
x² = 2f² + 6t
In this equation, x is the displacement, f is a constant, and t is the time. To find the velocity at any time t, we need to differentiate this equation with respect to time.
Finding the Velocity
1. Differentiate the Displacement Equation
We apply implicit differentiation to the equation x² = 2f² + 6t:
- Differentiate both sides with respect to time t.
- The left side becomes 2x(dx/dt), where dx/dt is the velocity (v).
- The right side, since 2f² is constant, becomes 6.
2. Setting Up the Equation
From the differentiation, we have:
- 2x(v) = 6
3. Solving for Velocity
To find the velocity (v), we rearrange the equation:
- v = 6 / (2x) = 3 / x
Final Result for Velocity
The velocity of the particle at any time t is given by:
- v = 3 / x
This means the velocity depends inversely on the displacement x. As the particle moves and changes its position, the velocity will vary according to this relationship.
Conclusion
Understanding the relationship between displacement and velocity is crucial in kinematics. In this case, the velocity decreases as the displacement increases, highlighting the nature of motion along the x-axis.