Question

A particle moves along a line with a velocity v(t)=−4t+4,

measured in meters per second. Find the total distance the particle
travels over the time interval [0,3]. Enter an exact answer.

Answer 1

The particle travels a total distance of 12 meters over the interval [0,3], determined by integrating its velocity function in sections where the velocity is positive and negative, and then summing these distances.

Step-by-step explanation:

Finding the Total Distance Traveled by a Particle

To find the total distance the particle travels over the time interval [0,3] with the velocity v(t) = -4t + 4, we look at the function and determine where the velocity changes sign, which is its zero. This occurs when v(t) = 0, or -4t + 4 = 0 which results in t = 1.

Since the total distance traveled is the integral of the absolute value of velocity over the given time period, we split the interval at t = 1. We then integrate the velocity function from 0 to 1 and from 1 to 3, ensuring we take the absolute value for each segment:

For t in [0,1], the velocity is positive, so we integrate as is:

Total distance from 0 to 1 = ∫(-4t + 4) dt

For t in [1,3], the velocity is negative, so we integrate the absolute value:

Total distance from 1 to 3 = ∫(4t - 4) dt

Performing the integrations:

From 0 to 1: (-4/2)t² + 4t | from 0 to 1 = (-2)(1²) + 4(1) = 2 meters

From 1 to 3: (4/2)t² - 4t | from 1 to 3 = [(2)(3²) - 4(3)] - [(2)(1²) - 4(1)] = 10 meters

The sum of these two distances gives us the total distance traveled by the particle from t = 0 to t = 3, which is 2 meters + 10 meters = 12 meters.

Answer 2

The total distance the particle travels over the time interval [0,3] is 6 meters.

To find the total distance the particle travels over the time interval [0,3], we need to calculate the area under the velocity-time graph. In this case, the velocity function is given as v(t) = -4t + 4.

To calculate the total distance traveled, we can break down the time interval into smaller intervals and calculate the distance traveled during each interval. We can then add up the distances to find the total distance.

Let's break down the interval [0,3] into smaller intervals: [0,1], [1,2], and [2,3].

For the interval [0,1], the velocity function is v(t) = -4t + 4. We can integrate this function to find the displacement during this interval:
∫(0 to 1) (-4t + 4) dt

Integrating, we get:
-2t^2 + 4t | (0 to 1)

Plugging in the values, we get:
[-2(1)^2 + 4(1)] - [-2(0)^2 + 4(0)]
[-2 + 4] - [0]
2

So, during the interval [0,1], the particle travels a distance of 2 meters.

Similarly, we can calculate the distances traveled during the intervals [1,2] and [2,3].

For the interval [1,2], the velocity function is still v(t) = -4t + 4. Integrating over this interval, we get a distance of 2 meters.

For the interval [2,3], the velocity function is v(t) = -4t + 4. Integrating over this interval, we get a distance of 2 meters.

Now, we can add up the distances traveled during each interval:
2 + 2 + 2 = 6