3.4 b) Particle Motion / Rectilinear Motion Our Lady of Lourdes Math Department

6.3 Rectilinear Motion 8. Apply derivatives 4 Solve a problem that involves applications and combinations of these skills. 3 Use implicit differentiation to find the derivative of an inverse function Use derivatives to solve problems involving rates of change with velocity, speed and acceleration Use derivatives to find absolute and relative extrema in real-life context (Unit 4 Ch 5-6) Use derivatives to model rates of change (Unit 4 Ch 5-6) 2 Uses implicit differentiation to find the derivative of an inverse but makes errors in writing the inverses or in computing the derivative Can state the relationship between position, velocity, speed and acceleration, and can write the velocity function given position, or acceleration function given the velocity function Shows understanding of using the derivatives to find extrema, but makes mistakes, or finds local extrema when absolute are required. Writes an equation to describe a situation but does not write a differential equation, or makes mistakes in evaluating the function. 1 With help, partial success at 2.0 & 3.0 content Even with help, no success

Vocabulary Day 1: Day 2: Day 3 Rectilinear Motion Velocity function Position function Velocity function Instantaneous velocity Day 2: Speed function Instantaneous speed Day 3 Acceleration Function Speeding up Slowing down

Rectilinear Motion Motion on a line Moving in a negative direction from the origin Moving in a positive direction from the origin

Position Function Horizontal axis: Vertical Axis: time position on a line Position function: s(t) s = position (s position) t = time s(t)= position changes as time changes Remember to mention that the “line” in rectllinear movement could also be vertical. When showing the sketchpad discuss how quickly the point moves on the line. Have students predict what part of the position graph will have the particle moving the fastest on the line. Moving in a negative direction from the origin Moving in a positive direction from the origin

Example 1 The figure below shows the position vs. time curve for a particle moving along an s-axis. In words, describe how the position of the particle is changing with time.

Example 1 The figure below shows the position vs. time curve for a particle moving along an s-axis. In words, describe how the position of the particle is changing with time. At t = 0, s(t) = -3. It moves in a positive direction until t = 4 and s(t) = 3. Then, it turns around and travels in the negative direction until t = 7 and s(t) = -1. The particle is stopped after that.

Example 1 Use the position and time graph to describe how the puppy was moving time position

Velocity Rate position change vs time change Velocity can be positive or negative positive: going in a positive direction negative: going in a negative direction Position Show the velocity sketchpad. Discuss the meaning of the slope of the position function Discuss instantaneous slope vs average slope. Show how the graph of velocity is related to graph of position. Velocity

Velocity function Velocity is the slope of the position function (change in position /change in time) velocity = Technically, this is instantaneous velocity Position Velocity Meaning Positive Slope Positive y’s moving in a positive direction Negative slope Negative y’s Moving in a negative direction

Velocity Rate at which a coordinate of a particle changes with time Instantaneous velocity s(t) = position with respect to time Instantaneous velocity at time t is: v(t) = positive – increasing slope – moving in a positive direction v(t) = negative – decreasing slope – moving in a negative direction

Practice Let s(t)= t3-6t2 be the position function of a particle moving along an s-axis were s is in meters and t is in seconds. Graph the position function On a number line, trace the path that the particle took. Where will the velocity be positive? Negative? Graph the instantaneous velocity. Identify on the velocity function when the particle was heading in a positive direction and when it was heading in a negative direction. Use Sketchpad Practice 1 to go over.

Velocity or Speed Speed: change in position with respect to time in any direction Velocity is the change in position with respect to time in a particular direction Thus – Speed cannot be negative – because going backwards or forwards is just a distance Thus – Velocity can be negative – because we care if we go backwards

Speed Absolute Value of Velocity example: 1) show speed sketchpad example: if two particles are moving on the same coordinate line with velocity of v=5 m/s and v=-5 m/s, then they are going in opposite directions but both have a speed of |v|=5 m/s

Example 2 Let s(t) = t3 – 6t2 be the position function of a particle moving along the s-axis, where s is in meters and t is in seconds. Find the velocity and speed functions, and show the graphs of position, velocity, and speed versus time.

Example 2 Let s(t) = t3 – 6t2 be the position function of a particle moving along the s-axis, where s is in meters and t is in seconds. Find the velocity and speed functions, and show the graphs of position, velocity, and speed versus time.

Example 2 The graphs below provide a wealth of visual information about the motion of the particle. For example, the position vs. time curve tells us that the particle is on the negative side of the origin for 0 < t < 6, is on the positive side of the origin for t > 6 and is at the origin at times t = 0 and t = 6.

Example 2 The velocity vs. time curve tells us that the particle is moving in the negative direction if 0 < t < 4, and is moving in the positive direction if t > 4 and is stopped at times t = 0 and t = 4 (the velocity is zero at these times). The speed vs. time curve tells us that the speed of the particle is increasing for 0 < t < 2, decreasing for 2 < t < 4 and increasing for t > 4.

Example - s(t)= t3-6t2 position time time velocity speed time

Practice Graph the velocity function What will the speed function look like? At what time(s) was the particle heading in a negative direction? Positive direction? 1) Practice 2 is on G. Sketchpad

Acceleration The rate at which the instantaneous velocity of a particle changes with time is called instantaneous acceleration. We define this as:

Acceleration The rate at which the instantaneous velocity of a particle changes with time is called instantaneous acceleration. We define this as: We now know that the first derivative of position is velocity and the second derivative of position is acceleration.

Example 3 Let s(t) = t3 – 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the acceleration function a(t) and show that graph of acceleration vs. time.

Example 3 Let s(t) = t3 – 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the acceleration function a(t) and show that graph of acceleration vs. time.

Speeding Up and Slowing Down We will say that a particle in rectilinear motion is speeding up when its speed is increasing and slowing down when its speed is decreasing. In everyday language an object that is speeding up is said to be “accelerating” and an object that is slowing down is said to be “decelerating.” Whether a particle is speeding up or slowing down is determined by both the velocity and acceleration.

The Sign of Acceleration A particle in rectilinear motion is speeding up when its velocity and acceleration have the same sign and slowing down when they have opposite signs.

Acceleration the rate at which the velocity of a particle changes with respect to time. If s(t) is the position function of a particle moving on a coordinate line, then the instantaneous acceleration of the particle at time t is or 1) looking at the velocity graph of sketcphad, when is the velocity changing fastest, slowest? not at all?

Example Let s(t) = t3 – 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the instantaneous acceleration a(t) and shows the graph of acceleration verses time 1)

Day 3: Speeding Up & Slowing down Speeding up when slope of speed is positive Slowing down when slope of speed is negative See acceleration sketch

Example When is s(t) speeding up and slowing down? position velocity time time velocity speed acceleration

Velocity & Acceleration function Slowing down Speeding up Slowing down Speeding up Velocity + Velocity - Velocity - Velocity + Rule: same sign speeding up, different sign slowing down Acceleration - Acceleration - Acceleration + Acceleration +

Analyzing Motion Graphically Algebraically Meaning Position Velocity  Acceleration Positive “s” values Positive side of the number line Negative side of the number line Negative “s” values s(t)=velocity. Look for Critical Pts Postive “v” values Moving in + direction 0 “v” values (CP) Turning/stopped Negative “v” values Moving in a – direction v(t)=acceleration + a, + v = speeding up - a, - v = speeding up + a, - v = slowing down - a, + v = slowing down Look for Critical Pts

Example Graphically Algebraically Meaning  Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3-21t2+60t+3 Analyze the motion of the particle for t>0 Example Graphically Algebraically Meaning  Always on postive side of number line Position Never 0 (t>0), always positive 0<t<2 going pos direction t=2 turning 2<t<5 going neg. direction + - + Velocity t=5 turning 2 5 t>5 going pos. direction t=5 t=2 t=0 0<t<2 slowing down 2<t<3.5 speeding up 2 5 3.5 v a - - + + 3.5<t<5 slowing down Acceleration + - - + 5<t speeding up

Example Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3 - 21t2 + 60t + 3 Analyze the motion of the particle for t>0 position velocity Acceleration

Position Direction of motion t --- v(t) positive direction negative +++ +++ 2 5 +++ --- +++ a(t) ----------- ++++++ 2 5 7/2 slowing down speeding up slowing down speeding up stop stop position velocity Acceleration

Day 4: Applications; Gravity s = position (height) s0= initial height v0= initial velocity t = time g= acceleration due to gravity g=9.8 m/s2 (meters and seconds) g=32 ft/s2 (feet and seconds) s0

Day 4: Applications; Gravity at time t= 0 an object at a height s0 above the Earth’s surface is given an upward or downward velocity of v0 and moves vertically (up or down) due to gravity. If the positive direction is up and the origin is the surface of the earth, then at any time t the height s=s(t) of the object is : g= acceleration due to gravity g=9.8 m/s2 (meters and seconds) g=32 ft/s2 (feet and seconds) s axis s0

Example Nolan Ryan was capable of throwing a baseball at 150ft/s (more than 102 miles/hour). Could Nolan Ryan have hit the 208 ft ceiling of the Houston Astrodome if he were capable of giving the baseball an upward velocity of 100 ft/s from a height of 7 ft? the maximum height occurs when velocity = 0 t=100/32=25/8 seconds s(25/8)=163.25 feet