if the position of a particle along y-axis is represented as a function of time t by the equation y(t)=t³ then find the displacement of the particle during the period t to t+​delta t

To find the displacement of a particle along the y-axis represented by the function \( y(t) = t^3 \) during the time interval from \( t \) to \( t + \Delta t \), we need to calculate the difference in the position of the particle at these two time points. Displacement is defined as the change in position, which can be mathematically expressed as:

Calculating Displacement

The displacement \( \Delta y \) over the interval from \( t \) to \( t + \Delta t \) can be found using the formula:

• Displacement, \( \Delta y = y(t + \Delta t) - y(t) \)

Now, we will evaluate \( y(t + \Delta t) \) and \( y(t) \):

Step 1: Evaluate \( y(t) \)

Substituting \( t \) into the function:

• \( y(t) = t^3 \)

Step 2: Evaluate \( y(t + \Delta t) \)

Next, we substitute \( t + \Delta t \) into the function:

• \( y(t + \Delta t) = (t + \Delta t)^3 \)

Now, we can expand \( (t + \Delta t)^3 \) using the binomial theorem:

• \( (t + \Delta t)^3 = t^3 + 3t^2(\Delta t) + 3t(\Delta t)^2 + (\Delta t)^3 \)

Step 3: Calculate the Displacement

Now that we have both \( y(t) \) and \( y(t + \Delta t) \), we can find the displacement:

• \( \Delta y = y(t + \Delta t) - y(t) \)
• \( \Delta y = \left( t^3 + 3t^2(\Delta t) + 3t(\Delta t)^2 + (\Delta t)^3 \right) - t^3 \)
• \( \Delta y = 3t^2(\Delta t) + 3t(\Delta t)^2 + (\Delta t)^3 \)

Final Expression for Displacement

The final expression for the displacement of the particle during the time interval from \( t \) to \( t + \Delta t \) is:

• \( \Delta y = 3t^2(\Delta t) + 3t(\Delta t)^2 + (\Delta t)^3 \)

This expression gives us a clear understanding of how the displacement depends on both the initial time \( t \) and the small increment \( \Delta t \). Each term in the expression represents different contributions to the total displacement, with the first term being the most significant for small \( \Delta t \). As \( \Delta t \) approaches zero, the linear term \( 3t^2(\Delta t) \) dominates, indicating that the particle's motion is primarily influenced by its velocity at that instant.