Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.

v(t) = 2t + 4; s(0) = 0

검증된 단계별 안내

Step 1: Recall that the position function s(t) is the integral of the velocity function v(t). To find s(t), we need to compute the indefinite integral of v(t) = 2t + 4.

Step 2: Set up the integral: \( s(t) = \int v(t) \, dt = \int (2t + 4) \, dt \). Break the integral into two parts: \( \int 2t \, dt \) and \( \int 4 \, dt \).

Step 3: Compute the integral of each term. For \( \int 2t \, dt \), use the power rule \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \). For \( \int 4 \, dt \), treat 4 as a constant and integrate to get \( 4t \).

Step 4: Combine the results of the integration: \( s(t) = t^2 + 4t + C \), where C is the constant of integration.

Step 5: Use the initial condition \( s(0) = 0 \) to solve for C. Substitute \( t = 0 \) into \( s(t) = t^2 + 4t + C \), which gives \( 0 = 0^2 + 4(0) + C \). Solve for C to find its value, and substitute it back into the position function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Velocity and Position Functions

Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function. In this context, the velocity function v(t) = 2t + 4 describes how the object's speed changes over time. To find the position function, we need to integrate the velocity function.

Integration

Integration is the process of finding the antiderivative of a function, which allows us to determine the original function from its rate of change. In this case, integrating the velocity function v(t) will yield the position function s(t). The integration process will also include a constant of integration, which can be determined using the initial position condition.

Initial Conditions

Initial conditions are specific values that help determine the constants in a function after integration. In this problem, the initial position s(0) = 0 provides a boundary condition that allows us to solve for the constant of integration after finding the position function. This ensures that the solution accurately reflects the object's position at the start of the observation.

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.v(t) = 2t + 4; s(0) = 0