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.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ 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 time interval \([t_1, t_2]\) is found by integrating the velocity function over that interval: \(\Delta s = \int_{t_1}^{t_2} v(t) \, dt\).

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

Evaluate the definite integral by computing \(\left[4.9t^2 - 3t\right]_{1}^{3} = (4.9 \times 3^2 - 3 \times 3) - (4.9 \times 1^2 - 3 \times 1)\) to find the displacement.

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주요 개념

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

Velocity and Displacement Relationship

Velocity is the rate of change of displacement with respect to time. Displacement over a time interval can be found by integrating the velocity function over that interval, which accumulates the total change in position.

Definite Integration

Definite integration calculates the net area under a curve between two limits. In this context, integrating the velocity function from t = 1 to t = 3 gives the total displacement during that time period.

Initial Conditions and Antiderivatives

An antiderivative of velocity gives the displacement function plus a constant. Using the initial condition s = s₀ at t = 0 allows determination of this constant, enabling calculation of displacement at any time.

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.iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.