what is total distance travelled given velocity equation v(t)=8-2t

To find the total distance traveled given the velocity equation \( v(t) = 8 - 2t \), we need to integrate the absolute value of the velocity function over a given time interval. Here's the process:

### Step 1: Find the time when the velocity becomes zero.
The total distance traveled depends on when the object changes direction, which occurs when the velocity equals zero.

\[
v(t) = 8 - 2t = 0
\]

Solving for \( t \):

\[
8 = 2t \quad \Rightarrow \quad t = 4
\]

This means that at \( t = 4 \), the velocity changes direction (from positive to negative).

### Step 2: Split the integral at \( t = 4 \).
Since the object changes direction at \( t = 4 \), we need to calculate the distance traveled from \( t = 0 \) to \( t = 4 \) and from \( t = 4 \) onward. We'll compute the distance separately for these intervals.

1. **For \( t \in [0, 4] \)**, the velocity is positive, so:
   \[
   \text{Distance}_1 = \int_0^4 (8 - 2t) \, dt
   \]

2. **For \( t > 4 \)**, the velocity is negative, so we take the absolute value:
   \[
   \text{Distance}_2 = \int_4^T |8 - 2t| \, dt
   \]
   (where \( T \) is the final time, but for a specific value of \( T \), you will need to substitute it).

### Step 3: Calculate the integrals.

1. **First integral (from 0 to 4):**
   \[
   \int_0^4 (8 - 2t) \, dt = \left[ 8t - t^2 \right]_0^4 = (8(4) - (4)^2) - (8(0) - (0)^2) = 32 - 16 = 16
   \]
   So, the distance traveled from \( t = 0 \) to \( t = 4 \) is 16 units.

2. **Second integral (from 4 to \( T \)):**
   For any specific \( T > 4 \), you would compute the integral of \( |8 - 2t| \) or equivalently \( -(8 - 2t) \).

If you are only interested in the distance up to \( t = 4 \), the total distance traveled is 16 units.

### Summary:
- Up to \( t = 4 \), the total distance traveled is 16 units.
- If you need the distance beyond \( t = 4 \), specify the time interval, and I'll compute the full distance traveled.

Do you want further details or have any questions?

Here are 5 related questions:
1. How would the total distance change if the initial velocity was different?
2. What is the object's acceleration at any given time?
3. How do you calculate displacement, and how is it different from total distance?
4. What happens to the total distance if \( v(t) \) is negative for all \( t \)?
5. Can you find the position function \( s(t) \) from this velocity equation?

**Tip**: Total distance accounts for the absolute value of the velocity; displacement only depends on the net change in position.

Find the total distance traveled given the velocity equation v(t) = 8 - 2t. This problem covers key calculus concepts, including integration of absolute value functions and kinematics. Suitable for advanced high school or early college students.

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