https://file.aiquickdraw.com/mathbot/file/17273109316411oy8zub8.jpg

Let's break down the two problems you have shared:

### 35. Average velocity
The velocity \( v(t) \) is given as \( v(t) = t^2 + 3t \) over the time interval \([0, 6]\).

**To find the average velocity:**

The average velocity is given by the formula:

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

where \( a = 0 \), \( b = 6 \), and \( v(t) = t^2 + 3t \).

So, the formula becomes:

\[
\text{Average velocity} = \frac{1}{6 - 0} \int_0^6 (t^2 + 3t) \, dt
\]

Let me calculate this for you.

### 36. Average velocity of a rock
The position \( s(t) \) is given by the equation \( s(t) = -16t^2 + 64 \). You are asked to find the average velocity of the rock during its flight.

**To find the average velocity:**

We use the formula:

\[
\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]

where \( s(t) = -16t^2 + 64 \), \( t_1 \) is the time at launch, and \( t_2 \) is the time when the rock hits the ground.

1. At \( t_1 = 0 \), \( s(0) = 64 \) feet (the initial height).
2. To find \( t_2 \), set \( s(t) = 0 \) to determine when the rock hits the ground:

\[
-16t^2 + 64 = 0
\]

Let me solve both these steps for you.Here are the solutions for the problems:

### Problem 35:
The **average velocity** of the object over the time interval \([0, 6]\) is:
\[
\text{Average velocity} = 21 \, \text{m/s}
\]

### Problem 36:
The **average velocity** of the rock during its flight (from launch to when it hits the ground) is:
\[
\text{Average velocity} = -32 \, \text{ft/s}
\]
(The negative sign indicates that the average motion is downward.)

Would you like more details on any of these steps or have further questions?

---

Here are 5 related questions to expand your understanding:
1. How do you interpret the physical meaning of average velocity when it is negative?
2. How would the process change if you were given a non-quadratic velocity function?
3. Can you find the total distance traveled by the object in problem 35 using calculus?
4. What happens if we extend the time interval beyond 6 seconds for problem 35?
5. How would the velocity function change for the rock in problem 36 if air resistance was considered?

**Tip:** Always ensure to correctly identify the time intervals when calculating average velocity. The interpretation of the results depends heavily on the limits chosen!

35. Average velocity The velocity in m/s of an object moving along a line over the time interval [0, 6] is v(t) = t^2 + 3t. Find the average velocity of the object over this time interval.
36. Average velocity A rock is launched vertically upward from the ground with a speed of 64 ft/s. The height of the rock (in ft) above the ground after t seconds is given by the function s(t) = -16t^2 + 64. Find its average velocity during its flight.

Learn how to calculate average velocity in two physics problems: one involving a quadratic velocity function and another involving vertical motion with a rock. These problems involve calculus and kinematic principles, ideal for high school students in grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation