Exploring Motion in One Dimension: Position, Velocity, and Acceleration

The Physics Problem: Analyzing Particle Movement Along the X-Axis

Question:

A particle moves along the x-axis with its position x given by the equation x = 12t² - 2t³. The task is to determine the particle's position, velocity, and acceleration at t = 3 seconds.

Solution: Calculating Position, Velocity, and Acceleration

To solve this problem, we use derivatives to find the velocity and acceleration from the given position function:

Phase 1 - Finding the Position

• The position of the particle at t = 3 seconds is found by substituting t = 3 into the equation x = 12t² - 2t³.

Phase 2 - Calculating Velocity

• Velocity (v) is the first derivative of position with respect to time (t). Hence, v = dx/dt.
• The velocity equation becomes v = 24t - 6t².
• Substitute t = 3 to find the velocity at that instant.

Phase 3 - Determining Acceleration

• Acceleration (a) is the second derivative of position with respect to time, or the first derivative of velocity. Therefore, a = d²x/dt² or dv/dt.
• The acceleration equation is a = 24 - 12t.
• Again, substitute t = 3 to find the acceleration at that moment