📐 Part 1: Mathematical Foundations

1. Which of the following best describes the relationship between position, velocity, and acceleration in calculus-based physics?

Velocity is the integral of position; acceleration is the integral of velocity

Velocity is the derivative of position; acceleration is the derivative of velocity

Position is the derivative of velocity; velocity is the derivative of acceleration

All three quantities are independent of each other

2. If position is given by x(t) = 3t² + 2t + 1, find the velocity function v(t) by taking the derivative:

3. Using your answer from question 2, find the acceleration function a(t):

🎯 Part 2: Conceptual Understanding

4. Check all statements that are true about one-dimensional motion (multiple selections allowed):

Displacement can be found by integrating velocity over time

Acceleration is always positive when an object is speeding up

The slope of a position vs. time graph gives velocity

The area under a velocity vs. time graph gives displacement

Velocity and speed are always the same value

5. Explain in your own words why acceleration is the derivative of velocity:

📊 Part 3: Graphical Analysis

6. Draw a sketch showing the relationship between position, velocity, and acceleration graphs for an object moving with constant acceleration:

7. If an object starts from rest and accelerates at 2 ft/s² for 5 seconds, what is its final velocity?

8. Using the scenario from question 7, how far does the object travel during those 5 seconds?

🔧 Part 4: Real-World Applications

9. Fill in the blanks about engineering applications:

In automotive engineering, analyzing vehicle _____________ profiles helps engineers design safer braking systems. The _____________ of velocity with respect to time gives us acceleration, which is crucial for understanding forces acting on passengers during sudden stops.

10. Match each physics concept with its real-world engineering application:

1. Displacement

2. Velocity

3. Acceleration

A. Force calculations in crash testing

B. GPS navigation distance calculations

C. Speed limit design for highways

11. A projectile is launched vertically upward. Describe what happens to its velocity and acceleration during flight: