13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


c. Find the distance traveled over the given interval.


v(t) = 4t³ - 24t²+20t on [0, 5]

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the total distance traveled by the object over the interval [0, 5]. Distance is calculated by integrating the absolute value of the velocity function v(t) over the given interval.
2. Step 2: Identify the velocity function v(t) = 4t³ - 24t² + 20t and the interval [0, 5]. To compute the distance, we need to account for any changes in direction (where v(t) = 0) because the absolute value of velocity is required.
3. Step 3: Solve for the critical points where v(t) = 0. Set 4t³ - 24t² + 20t = 0 and factorize the equation: t(4t² - 24t + 20) = 0. Solve for t to find the points where the velocity changes sign within the interval [0, 5].
4. Step 4: Determine the sign of v(t) in each subinterval created by the critical points. This involves testing the sign of v(t) in intervals such as [0, t₁], [t₁, t₂], ..., [tₙ, 5], where t₁, t₂, ..., tₙ are the critical points. Use these signs to apply the absolute value of v(t) in the integration.
5. Step 5: Compute the total distance traveled by integrating |v(t)| over each subinterval. For each subinterval, integrate the absolute value of v(t) and sum the results to find the total distance. Use the formula: ∫|v(t)| dt from t=a to t=b for each subinterval.

Key Concepts

Velocity Function

The velocity function v(t) describes the rate of change of an object's position with respect to time. In this case, v(t) = 4t³ - 24t² + 20t is a polynomial function that indicates how the object's speed varies over time. Understanding this function is crucial for determining how far the object travels during a specific time interval.

Definite Integral

To find the distance traveled over a time interval, we use the definite integral of the velocity function. The integral calculates the net area under the velocity curve from the start to the end of the interval, which corresponds to the total displacement. In this problem, we will evaluate the integral of v(t) from t = 0 to t = 5.

Distance vs. Displacement

While displacement refers to the change in position from the start to the end of the interval, distance accounts for the total path traveled, regardless of direction. If the velocity function changes sign within the interval, it indicates that the object may reverse direction, which affects the total distance calculation. Thus, it is important to analyze the velocity function for any intervals where it may be negative.