A particle P is projected from the origin O so that it moves along the x-axis. At time t after projection, the velocity of the particle v is given by v = 3t^2 - 24t + 45.) Find an expression for the displacement of P in the first 3 seconds of its motion using calculus (integration).

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Physics

Merlin

14 February, 13:09

A particle P is projected from the origin O so that it moves along the x-axis. At time t after projection, the velocity of the particle v is given by v = 3t^2 - 24t + 45.) Find an expression for the displacement of P in the first 3 seconds of its motion using calculus (integration).

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Aimee Bauer · Answer 1

x = t³ - 12t² + 45t

Explanation:

Position is the integral of velocity.

x = ∫ v dt

x = ∫ (3t² - 24t + 45) dt

x = t³ - 12t² + 45t + C

The particle is originally at the origin, so at t = 0, x = 0.

0 = (0) ³ - 12 (0) ² + 45 (0) + C

0 = C

x = t³ - 12t² + 45t