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 ?

Verified step by step guidance

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

Key Concepts

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.