How to Find the Acceleration Given the Speed and Position as Vector Valued Functions | Summary and Q&A

TL;DR

Finding the velocity and position vectors by integrating acceleration and using initial conditions.

Key Insights

• 🧘 Acceleration is integrated to find the velocity vector, while the velocity vector is integrated to find the position vector.
• 🧘 Initial conditions are used to determine the constant vector in the velocity and position vectors.
• ✖️ The problem involves basic vector operations, such as addition, scalar multiplication, and integration.
• 👌 The I, J, and K unit vectors are used to represent vector components in different directions.
• 👻 Substituting specific values into vector equations allows us to determine the velocity and position at those times.
• ✊ The power rule is used to integrate terms involving T raised to a power.
• 🧘 The final position vector represents the position of an object at a specific time.

Transcript

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Questions & Answers

Q: How do we find the velocity vector (V)?

By integrating the given acceleration vector, we obtain V = 8T in the I direction, 4 in the J direction, and 6T in the K direction, with a constant vector of 4J.

Q: How is the position vector (R) obtained?

To find R, we integrate the velocity vector. The resulting position vector is R = (4T^2/2) in the I direction, 4T in the J direction, and (3T^2/2) in the K direction, with no constant vector added.

Q: What are the initial conditions used in the problem?

The initial condition for the velocity vector is V(0) = 4J, which allows us to determine the constant vector C as 4J. The initial condition for the position vector is R(0) = 0, which helps find C as the zero vector.

Q: How do we evaluate R at T = 9?

To find R(9), we substitute T = 9 into the position vector equation. After calculations, we get R(9) = 324I + 36J + 243K, representing the position vector at 9 seconds.

Summary & Key Takeaways

• The problem involves finding the velocity (V) and position (R) vectors at a given time (T) using the given acceleration (a) and initial conditions.

• The acceleration is integrated to obtain the velocity vector (V), which consists of three components: 8T in the I direction, 6T in the K direction, and a constant vector C in the J direction.

• The velocity vector is then integrated to obtain the position vector (R), which consists of three components: (4T^2/2) in the I direction, (4T) in the J direction, and (3T^2/2) in the K direction.