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) = cos2t; v(0) = 5; s(0) = 7

검증된 단계별 안내

Identify the given acceleration function: \(a(t) = \cos^2(t)\), the initial velocity \(v(0) = 5\), and the initial position \(s(0) = 7\).

Recall that velocity is the integral of acceleration with respect to time: \(v(t) = v(0) + \int_0^t a(u) \, du\).

Set up the integral for velocity: \(v(t) = 5 + \int_0^t \cos^2(u) \, du\). To solve this integral, use the trigonometric identity \(\cos^2(u) = \frac{1 + \cos(2u)}{2}\) to simplify the integrand.

Once you find \(v(t)\), find the position function by integrating velocity: \(s(t) = s(0) + \int_0^t v(u) \, du\).

Set up the integral for position: \(s(t) = 7 + \int_0^t v(u) \, du\). After computing this integral, you will have the position function \(s(t)\).

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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 derivative of velocity with respect to time, and velocity is the derivative of position. To find velocity from acceleration, integrate the acceleration function, and to find position from velocity, integrate the velocity function. 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 given its derivative by integrating, and use initial values to solve for constants, which is essential for finding velocity and position from acceleration.

Initial Conditions and Constants of Integration

When integrating acceleration to find velocity, and velocity to find position, each integration introduces an unknown constant. Initial conditions like v(0) and s(0) provide specific values to solve for these constants, ensuring the solution matches the physical scenario described.

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) = cos2t; v(0) = 5; s(0) = 7