If S=2t+1 describe the law of motion of a body where S is the distance and t is time using definite integral technique find average velocity between t=0 and t=3 seconds 

We are given the equation \( S = 2t + 1 \), where \( S \) is the distance traveled and \( t \) is the time. To find the average velocity of the body between \( t = 0 \) and \( t = 3 \), we can use the following formula for average velocity:

\[
\text{Average Velocity} = \frac{\text{Total Distance Traveled}}{\text{Total Time Interval}}
\]

Alternatively, using definite integrals, we can express the average velocity over a time interval \( [a, b] \) as:

\[
\text{Average Velocity} = \frac{1}{b - a} \int_{a}^{b} v(t) \, dt
\]

### Step 1: Find the Velocity Function
The velocity \( v(t) \) is the derivative of the distance function \( S(t) \) with respect to time \( t \):

\[
v(t) = \frac{dS}{dt} = \frac{d}{dt} (2t + 1) = 2
\]

This means that the velocity is constant at 2 m/s.

### Step 2: Set up the Definite Integral
Since \( v(t) = 2 \), the integral over the interval \( [0, 3] \) becomes:

\[
\int_{0}^{3} 2 \, dt
\]

### Step 3: Solve the Integral
We can compute the definite integral as follows:

\[
\int_{0}^{3} 2 \, dt = 2t \Big|_0^3 = 2(3) - 2(0) = 6
\]

### Step 4: Calculate the Average Velocity
Since the velocity is constant, the average velocity over the interval is simply:

\[
\text{Average Velocity} = \frac{1}{3 - 0} \int_{0}^{3} 2 \, dt = \frac{1}{3} \times 6 = 2 \, \text{m/s}
\]

Thus, the average velocity between \( t = 0 \) and \( t = 3 \) seconds is \( 2 \, \text{m/s} \).

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Would you like a more detailed explanation of any part of this? Here are some related questions:

1. How do you find average velocity if the velocity is not constant?
2. What is the significance of taking the derivative of a distance function?
3. Can we apply this method to non-linear distance functions?
4. How would the result change if we had an acceleration term?
5. What is the relationship between definite integrals and physical quantities like distance or velocity?

**Tip:** The derivative of the position function gives the velocity, and the derivative of the velocity gives the acceleration!

If S = 2t + 1 describe the law of motion of a body where S is the distance and t is time using definite integral technique find average velocity between t = 0 and t = 3 seconds

Learn how to calculate the average velocity of a body using definite integrals for the motion equation S = 2t + 1. This solution covers key calculus concepts such as velocity functions, integral techniques, and kinematics, suitable for high school students in Grades 11-12.

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