91Ó°ÊÓ

In kinematics, displacement is defined as the change in position of an object over a certain period of time. It helps us understand how far the object has moved from its initial position. Calculating displacement requires careful integration of the velocity function, which describes the rate of change of position with respect to time.

To find the displacement, we integrate the velocity function with respect to time. This is because displacement is the area under the velocity-time graph. Hence, in mathematics, we define displacement as the integral of velocity over the time interval considered.

If you have a velocity function given as a function of time, for example, like in our problem: \( v = v_0 + g t + f t^2 \), you compute the displacement \( x(t) \) by integrating this function of time. Doing so gives the equation \( x(t) = \int (v_0 + g t + f t^2) \, dt \), which leads to a structured formula for displacement when solved thoroughly.

Integration is a fundamental concept in calculus, and it plays a significant role in kinematics, especially for calculating displacement. The velocity of an object functions as a derivative of displacement with respect to time. By integrating the velocity function, we work backwards to find the original position function or displacement.

For our equation \( v = v_0 + g t + f t^2 \), we perform integration to determine how the position \( x \) changes with time. Integrating each term separately gives:

• \( v_0 t \) — since \( v_0 \) is a constant, integrating \( v_0 \) with respect to \( t \) gives \( v_0 t \)
• \( \frac{g t^2}{2} \) — the integral of \( g t \) is \( \frac{g t^2}{2} \)
• \( \frac{f t^3}{3} \) — integrating \( f t^2 \) results in \( \frac{f t^3}{3} \)

This yields the displacement function:
\( x(t) = v_0 t + \frac{g t^2}{2} + \frac{f t^3}{3} + C \), where \( C \) is the constant of integration that depends on initial conditions.

Initial conditions serve as crucial pieces of information when solving kinematic equations. They help us determine unknown constants that arise from the integration process. In our specific exercise, the initial condition provided is \( x = 0 \) when \( t = 0 \). This condition essentially tells us the starting position of the particle.

When integrating a velocity function, we find an integration constant \( C \) which is unknown. We use these initial conditions to solve for \( C \), ensuring that the displacement function accurately reflects the motion of the particle starting from the given initial values.

For instance, with \( t = 0 \) and \( x = 0 \), substituting these values into the equation \( x(t) = v_0 t + \frac{g t^2}{2} + \frac{f t^3}{3} + C \) gives us \( 0 = C \). Hence, the constant does not affect the displacement formula for further time calculations. This step is critical for obtaining the correct solution when analyzing particle motion in time and ensures our displacement results accurately reflect the given scenario.