An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.
a. Find the velocity and position for t≥0.

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Start with the given acceleration function: \(a(t) = 2 \sin\left(\frac{\pi t}{4}\right)\). Recall that acceleration is the derivative of velocity, so \(a(t) = v'(t)\).

Integrate the acceleration function to find the velocity function: \(v(t) = \int a(t) \, dt = \int 2 \sin\left(\frac{\pi t}{4}\right) dt\). Remember to include the constant of integration \(C_1\).

Use the initial condition for velocity, \(v(0) = -\frac{8}{\pi}\), to solve for the constant \(C_1\) after integrating.

Next, integrate the velocity function to find the position function: \(s(t) = \int v(t) \, dt\). Again, include a constant of integration \(C_2\).

Use the initial position condition \(s(0) = 0\) to solve for the constant \(C_2\). This will give you the complete expressions for velocity \(v(t)\) and position \(s(t)\) for \(t \geq 0\).

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주요 개념

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

Relationship Between Acceleration, Velocity, and Position

Acceleration is the derivative of velocity with respect to time, and velocity is the derivative of position. To find velocity and position from acceleration, you integrate acceleration once to get velocity and integrate velocity to get position, applying initial conditions to determine constants of integration.

Integration of Trigonometric Functions

Integrating functions like sin(kt) involves using standard integral formulas, such as ∫sin(kt) dt = -cos(kt)/k + C. Understanding how to handle the coefficient inside the sine function is essential to correctly find antiderivatives when solving for velocity and position.

Applying Initial Conditions

Initial conditions, such as v(0) and s(0), are used to find the constants of integration after integrating acceleration and velocity. These conditions ensure the solution matches the physical scenario and provide unique expressions for velocity and position.

An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0. a. Find the velocity and position for t≥0.