How to Calculate Velocity and Position from Acceleration

TL;DR

To find the velocity and position vectors from acceleration, integrate the acceleration function. Use initial conditions to determine any constants, resulting in expressions for velocity and position vectors that can then be evaluated at a specific time, such as T = 9.

Transcript

in this problem we have to find V of T which is the velocity R of T which is a position and evaluate R at T equals 9 here a is the acceleration and it's given in the problem along with these two conditions so solution so the first thing we'll do is we'll find V of T so whenever you have the acceleration and you integrate it you'll get the velocity ... Read More

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.

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.