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) = 2√t; s(0) = 1

검증된 단계별 안내

Step 1: Recall that the position function s(t) can be found by integrating the velocity function v(t) with respect to time t. The relationship is given by:

s(t) = ∫v(t) dt + C

, where C is the constant of integration.

Step 2: Substitute the given velocity function v(t) = 2√t into the integral. This gives:

s(t) = ∫2√t dt + C

Step 3: Simplify the integral. Recall that

√t

can be written as

t^{1/2}

. Using the power rule for integration,

∫t^n dt = (t^{n+1})/(n+1)

, apply this to integrate

2t^{1/2}

Step 4: After integrating, you will have an expression for s(t) in terms of t and the constant of integration C. Use the initial condition s(0) = 1 to solve for C. Substitute t = 0 into the position function and set s(0) equal to 1.

Step 5: Combine the result of the integration and the value of C to write the final position function s(t).

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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. To find the position function from a given velocity function, one must integrate the velocity function. This relationship is fundamental in calculus, as it connects the concepts of motion and change.

Integration

Integration is the process of finding the integral of a function, which essentially reverses differentiation. In the context of finding a position function from a velocity function, integration allows us to determine the accumulated position over time. The result of integration includes a constant of integration, which can be determined using initial conditions.

Initial Conditions

Initial conditions are specific values that help define a particular solution to a differential equation or integral. In this case, the initial position s(0) = 1 provides the necessary information to solve for the constant of integration after finding the position function. This ensures that the solution accurately reflects the scenario described in the problem.

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) = 2√t; s(0) = 1