29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = e^−t; v(0) = 60; s(0) = 40

검증된 단계별 안내

Identify the given acceleration function: \(a(t) = e^{-t}\), the initial velocity \(v(0) = 60\), and the initial position \(s(0) = 40\).

Recall that velocity is the integral of acceleration with respect to time: \(v(t) = \int a(t) \, dt + C_1\). Here, \(C_1\) is the constant of integration that we will find using the initial velocity.

Integrate the acceleration function: \(v(t) = \int e^{-t} \, dt + C_1\). The integral of \(e^{-t}\) is \(-e^{-t}\), so \(v(t) = -e^{-t} + C_1\).

Use the initial velocity condition \(v(0) = 60\) to solve for \(C_1\): substitute \(t=0\) into \(v(t)\) to get \(60 = -e^{0} + C_1\), then solve for \(C_1\).

Next, find the position function by integrating the velocity function: \(s(t) = \int v(t) \, dt + C_2\). Use the initial position \(s(0) = 40\) to solve for the constant of integration \(C_2\).

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주요 개념

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

Acceleration, Velocity, and Position Relationship

Acceleration is the rate of change of velocity with respect to time, and velocity is the rate of change of position. Given acceleration, velocity can be found by integrating acceleration, and position can be found by integrating velocity. Initial conditions help determine the constants of integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that integration can be reversed by differentiation. It allows us to find a function from its derivative by integrating, and use initial values to solve for constants, which is essential when finding velocity and position from acceleration.

Initial Conditions in Differential Equations

Initial conditions specify the value of a function at a particular point, enabling the determination of integration constants after integrating. For motion problems, initial velocity and position are used to find the exact velocity and position functions from their derivatives.

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).a(t) = e^−t; v(0) = 60; s(0) = 40