Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.

(a) How far does the object travel, for 0 ≤ t ≤ 4 ?

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Step 1: Recognize that the displacement of the object can be found by integrating the velocity function v(t) = 2t + 5 over the given time interval [0, 4]. The formula for displacement is: \( s(t) = \int v(t) \, dt \).

Step 2: Set up the definite integral for displacement: \( \int_{0}^{4} (2t + 5) \, dt \). This represents the total distance traveled by the object from t = 0 to t = 4.

Step 3: Break the integral into two parts for easier computation: \( \int_{0}^{4} 2t \, dt + \int_{0}^{4} 5 \, dt \).

Step 4: Compute each integral separately. For \( \int_{0}^{4} 2t \, dt \), use the power rule of integration: \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \). For \( \int_{0}^{4} 5 \, dt \), treat 5 as a constant and multiply it by the length of the interval.

Step 5: Add the results of the two integrals together to find the total displacement. This will give the total distance traveled by the object over the interval [0, 4].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity

Velocity is the rate of change of an object's position with respect to time. In this context, the velocity function v(t) = 2t + 5 describes how the object's speed changes over time. Understanding velocity is crucial for determining how far the object travels over a given time interval.

Displacement

Displacement refers to the change in position of an object and can be calculated as the integral of the velocity function over a specific time interval. In this case, to find the total distance traveled by the object from t = 0 to t = 4, we need to integrate the velocity function v(t) over that interval.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, over a specified interval. In this problem, we will use the definite integral of the velocity function from t = 0 to t = 4 to find the total distance traveled by the object, which is essential for solving the question.

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