Finding displacement from an antiderivative of velocity

a. Suppose that the velocity of a body moving along the s-axis is

ds/dt = v = 9.8t − 3.

i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

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Identify the given velocity function: \(v(t) = \frac{ds}{dt} = 9.8t - 3\).

Recall that displacement over the interval \([t_1, t_2]\) is given by the change in position \(s(t_2) - s(t_1)\), which can be found by integrating the velocity function over that interval.

Set up the definite integral for displacement from \(t=1\) to \(t=3\): \(\Delta s = \int_{1}^{3} (9.8t - 3) \, dt\).

Find the antiderivative (indefinite integral) of the velocity function: \(S(t) = \int (9.8t - 3) \, dt = 4.9t^2 - 3t + C\), where \(C\) is the constant of integration.

Use the initial condition \(s(0) = 5\) to solve for \(C\) by substituting \(t=0\) into \(S(t)\), then evaluate \(S(3) - S(1)\) to find the displacement over the interval \([1,3]\).

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주요 개념

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

Relationship Between Velocity and Displacement

Velocity is the rate of change of displacement with respect to time, expressed as v = ds/dt. To find displacement over a time interval, we integrate the velocity function over that interval, which gives the net change in position.

Definite Integration to Find Displacement

Definite integration of the velocity function from the initial time to the final time calculates the total displacement. This integral sums the instantaneous velocities over time, accounting for direction and magnitude of movement.

Initial Conditions and Antiderivatives

An antiderivative of velocity gives the displacement function s(t) up to a constant. Using the initial condition s(0) = 5 allows us to solve for this constant, ensuring the displacement function accurately reflects the body's position at all times.

Finding displacement from an antiderivative of velocitya. Suppose that the velocity of a body moving along the s-axis is ds/dt = v = 9.8t − 3.i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.