Question

What is the simplified form of the solution function \(v(t)\) obtained by evaluating the integral on the right, considering the initial value \(v(0) = 0\) in the given context?

Answer 1

The simplified form of the solution function \(v(t)\) with the initial value \(v(0) = 0\) is \(v(t) = \frac{1}{2}
`t^2`\).

Step-by-step explanation:

In the given context, the problem involves finding the solution function \(v(t)\) by evaluating the integral with the initial value \(v(0) = 0\). The integration of the function yields \(v(t) = \frac{1}{2}
`t^2` + C\), where \(C\) is the constant of integration. However, considering the initial value \(v(0) = 0\), we can determine that \(C = 0\), resulting in the simplified form \(v(t) = \frac{1}{2}
`t^2`\).

This simplification is achieved by substituting the initial value into the general solution and solving for the constant. In this case, the initial condition \(v(0) = 0\) helps determine that the constant \(C\) is zero, leading to the final simplified form of the solution. Therefore, the function \(v(t) = \frac{1}{2}
`t^2`\) describes the velocity in the given context, considering the initial condition.

The expression \(\frac{1}{2}
`t^2`\) represents the position function, and the initial value \(v(0) = 0\) ensures that the particle starts from rest at the origin. The constant acceleration, indicated by the term \(\frac{1}{2}
`t^2`\), captures the relationship between velocity and time.

Answer 2

Main Answer:

The simplified form of the solution function \(v(t)\) obtained by evaluating the integral on the right, considering the initial value \(v(0) = 0\) is \[v(t) =
`\int_(0)^(t) f(u) \, du\]`.

Step-by-step explanation:

In the context of this problem, we are dealing with a differential equation that involves the function \(v(t)\). The given solution function is expressed as an integral, specifically the definite integral from 0 to \(t\) of \(f(u)\) with respect to \(u\). This integral represents the accumulation of values of \(f(u)\) over the interval \([0, t]\), and it is subject to the initial condition \(v(0) = 0\).

The integral notation
`\(\int_(0)^(t) f(u) \, du\)` signifies the antiderivative of the function \(f(u)\) with respect to \(u\), evaluated from 0 to \(t\). The result is the net change in the antiderivative function over the interval \([0, t]\), which, in this case, represents the solution function \(v(t)\). The initial condition \(v(0) = 0\) provides the starting point for the accumulation process, ensuring that the integral starts from zero.

This expression encapsulates the solution to the differential equation and is simplified to this form, taking into account the initial condition. It provides a mathematical representation of how the function \(v(t)\) evolves over time based on the given conditions.