2 Position, Velocity and Acceleration All fall under rectilinear motionMotion along a straight lineWe are normally given a function relating the position of a moving object with respect to time.Velocity is the derivative of positionAcceleration is the derivative of velocity
3 Position, Velocity and Acceleration s(t) or x(t)Velocityv(t) or s’(t)Accelerationa(t) or v’(t) or s’’(t)Speed is the absolute value of velocity
4 ExampleIf 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 backwardsWhen velocity is positive, the particle is moving to the right or forwardsWhen velocity and acceleration have the same sign, the speed is increasingWhen 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 Changes direction when velocity = 0 and acceleration does not ExampleIf 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 Particle is slowing down when, ExampleUsing 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 4462Particle is slowing down when,0 < t < 24 < t < 6tv(t)+-a(t)
9 3 is not in our interval so it will not affect our problem! ExampleWhen velocity = 0When 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 intervalThe 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 Divide into intervals; 02 and 24 ExampleHow far does a particle travel between zero and four seconds if its position is given by:Divide into intervals; 02 and 24
11 Divide into intervals; 01 and 12 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 zeroC. Find the total distance traveled by the body from t = 0 to t = 2Velocity = 0 at 1!Divide into intervals; 01 and 12B. Find the body’s speed each time the acceleration is zero
12 Velocity increasing: (1, 2) and (3, ∞) At any time t, the position of a particle moving along an axis is:A. When is the body moving forward? backwards?13v(t)+-Forward (0, 1) and (3, ∞)Backwards from (1, 3)B. When is the velocity increasing? decreasing?123Velocity increasing: (1, 2) and (3, ∞)Velocity decreasing:(0, 1) and (2, 3)tv(t)+-a(t)