Position, Velocity and Acceleration

1. Position, Velocity and Acceleration 3.2

2. Position, Velocity and Acceleration • All fall under rectilinear motion • Motion along a straight line • We are normally given a function relating the position of a moving object with respect to time. • Velocity is the derivative of position • Acceleration is the derivative of velocity

3. Position, Velocity and Acceleration • Position • s(t) or x(t) • Velocity • v(t) or s’(t) • Acceleration • a(t) or v’(t) or s’’(t) • Speed is the absolute value of velocity

4. Example • If the position of a particle at time t is given by the equation below, find the velocity and acceleration of the particle at time, t = 5.

5. Position, Velocity and Acceleration • When velocity is negative, the particle is moving to the left or backwards • When velocity is positive, the particle is moving to the right or forwards • When velocity and acceleration have the same sign, the speed is increasing • When velocity and acceleration have opposite signs, the speed is decreasing. • When velocity = 0 and acceleration does not, the particle is momentarily stopped and changing direction.

7. Example • If the position of a particle is given below, find the point at which the particle changes direction. Changes direction when velocity = 0 and acceleration does not

8. Example • Using the previous function, find the interval of time during which the particle is slowing down. V(t) = 0 at 2 and 6, a(t) = 0 at 4 4 6 0 2 Particle is slowing down when, 0 < t < 2 4 < t < 6

9. Example When velocity = 0 When does this occur? • How far does a particle travel between the eighth and tenth seconds if its position is given by: To find the total distance we must find if the particle changes directions at any time in the interval The object may travel forward then backwards, thus s(10) – s(8) is really only the displacement not the total distance! 3 is not in our interval so it will not affect our problem!

10. Example • How far does a particle travel between zero and four seconds if its position is given by: Divide into intervals; 02 and 24

11. At any time t, the position of a particle moving along an axis is: A. Find the body’s acceleration each time the velocity is zero C. Find the total distance traveled by the body from t = 0 to t = 2 Velocity = 0 at 1! Divide into intervals; 01 and 12 B. Find the body’s speed each time the acceleration is zero

12. At any time t, the position of a particle moving along an axis is: A. When is the body moving forward? backwards? 1 3 Forward (0, 1) and (3, ∞) Backwards from (1, 3) B. When is the velocity increasing? decreasing? 0 1 2 3 Velocity increasing: (1, 2) and (3, ∞) Velocity decreasing: (0, 1) and (2, 3)