Particle Motion in Calculus

Particle Motion in Calculus

Read each question carefully and provide detailed answers, showing all your work where applicable. Remember the relationships between position, velocity, and acceleration.

1. If the position of a particle is given by s(t) = t^3 - 6t^2 + 9t, what is the velocity of the particle at t = 2?

2. When is the particle at rest if its velocity is v(t) = 3t^2 - 12t?

1. The derivative of the position function gives the   function.

2. If the velocity and acceleration have opposite signs, the particle is  .

1. A particle moves along a horizontal line with position function s(t) = t^3 - 3t^2 - 9t + 5, where t is in seconds and s is in meters.

a) Find the velocity function v(t).

b) Find the acceleration function a(t).

c) When is the particle moving to the right?

d) When is the particle speeding up?

1. If the velocity of a particle is constant, then its acceleration is zero.

2. A particle is moving to the left if its velocity is negative.

Consider a particle whose position is given by s(t) = t^2 - 4t + 3.

1. Draw a number line to represent the particle's motion. Indicate the direction of motion and when the particle is at rest.