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) = −32; v(0)=50; s(0)=0

검증된 단계별 안내

Identify the given acceleration function: \(a(t) = -32\), the initial velocity \(v(0) = 50\), and the initial position \(s(0) = 0\).

Recall that velocity is the integral of acceleration with respect to time: \(v(t) = \int a(t) \, dt + C_1\).

Integrate the acceleration function: \(v(t) = \int -32 \, dt = -32t + C_1\).

Use the initial velocity condition \(v(0) = 50\) to solve for the constant \(C_1\): substitute \(t=0\) into \(v(t)\) and set equal to 50.

Next, find the position function by integrating the velocity function: \(s(t) = \int v(t) \, dt + C_2\). Then use the initial position \(s(0) = 0\) to solve for \(C_2\).

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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, integrating it once with respect to time yields velocity, and integrating velocity gives position. 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, which is essential for finding velocity and position from acceleration.

Initial Conditions and Constants of Integration

When integrating acceleration to find velocity and position, constants of integration appear. Initial velocity and initial position values are used 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) = −32; v(0)=50; s(0)=0