Displacement Function
The displacement of the particle is given as:
x = a0 + (a1 * t) / 2 - (a2 * t^2) / 3
Where:
- a0, a1, and a2 are constants.
- t is time.
Velocity Calculation
To find the acceleration, we first need to determine the velocity by differentiating the displacement function with respect to time (t).
- Velocity (v) = dx/dt
- Differentiate x:
v = d/dt [a0 + (a1 * t) / 2 - (a2 * t^2) / 3]
- Resulting in:
v = (a1 / 2) - (2 * a2 * t) / 3
Acceleration Calculation
Next, we find the acceleration by differentiating the velocity function with respect to time (t).
- Acceleration (a) = dv/dt
- Differentiate v:
a = d/dt [(a1 / 2) - (2 * a2 * t) / 3]
- Resulting in:
a = - (2 * a2) / 3
Summary of Key Points
- The displacement function is a polynomial expression of time.
- Velocity is found by taking the first derivative of displacement.
- Acceleration is found by taking the derivative of velocity.
- The final expression for acceleration is a constant value of - (2 * a2) / 3.
This process illustrates the steps to determine acceleration from a given displacement function for a particle in motion.