Particle Motion Notes Position When an object moves, its position is a function of time. For its position function, we will denote the variable s(t). Example 1: For s( t) t t 3, show its position on the number line for t 0,1,,3,4. Velocity When an object moves, its position changes over time. So we can say that the velocity function, v(t) is the change of the position function over time. We know this to be the derivative, and can thus say that v(t)=s (t). For convenience sake, we will define v(t) in the following way: Motion v(t) > 0 v(t) < 0 v(t) = 0 Horizontal Line Object moves to the right Object moves to the left Object stopped Vertical Line Object moves up Object moves down Object stopped Speed is not synonymous with velocity. Upward motion is defined as positive velocity and downward motion is defined as negative velocity. Speed does not indicate direction. Se we define the speed function: Speed The speed of an object must either be positive or zero (meaning the object has stopped). Example : The graph shows the position, s(t), of a particle along a horizontal axis. (a) When is the particle moving to the right? Explain why. (b) When is the particle moving to the left? Explain why. (c) When is the particle standing still? Explain why. (d) Graph the particle s velocity and speed (where defined).
Acceleration The definition of acceleration is the change in velocity over time. We know this to be a derivative and can thus say that a(t)=v (t)=s (t). So given a position function s(t), we can now determine both the velocity and acceleration function. On your cars, you have two devices that change velocity. What are they? For convenience sake, let us define the acceleration function like this: Motion a(t) > 0 a(t) < 0 a(t) = 0 Horizontal Line Object accelerating to the right Object accelerating to the left Velocity not changing Vertical Line Object accelerating upwards Object accelerating downwards Velocity not changing Just because an object s acceleration is zero does not mean that the object is stopped. It means that the velocity is not changing. What device on your car will keep the car s acceleration equal to zero? Also, just because you have a positive acceleration doesn t mean that you are moving to the right. For instance, suppose you were walking to the right vt ( ) 0, when all of a sudden a large wind started to blow to the left at ( ) 0. What would that do to your velocity? The Relationship Between Velocity and Acceleration Fill in each box with either of the phrases: speeding up, slowing down, constant speed, or stopped. How are we moving? a(t) > 0 a(t) < 0 a(t) = 0 v(t) > 0 v(t) < 0 v(t) = 0 Example 3: The graph shows the velocity v = f (t) of a particle moving along a horizontal coordinate axis. (a) When does the particle reverse direction? Explain why. (b) When is the particle moving at a constant speed? Explain why. (c) When is the particle moving at its greatest speed? (d) Graph the acceleration.
3 Example 4: A particle is moving along a horizontal line with position function s( t) t 9t 4t 4. Do an analysis of the particle s direction, acceleration, motion (speeding up or slowing down), and position. When Tips an to object Solve is subjected a Particle to gravity, Motion its position Analysis function Problem is given by, where Step 1: Find is measured v(t). Solve in seconds, for v(t)=0. is measured in feet, is the Step : Make a number line of v(t) showing when the object is stopped and the sign and direction of initial velocity (velocity at =0) and is the initial position (position at =0). the object at times to the left and right of that. Assume t > 0. The Step formula 3: Find is a(t). given Solve by for a(t)=0. Step 4: Make a number line of a(t) showing when the object has a positive and negative acceleration. if is Scale measured it exactly in meters. like the v(t) number line. Step 5: Make a motion line directly below the last two lines putting all critical values, multiplying the signs and interpreting according to the chart you completed at the top of this page. Step 6: Make a position graph to show where the object is at critical times and how it moves. 3 Example 5: A particle is moving along a horizontal line with position function s( t) t 6t 9t 1. Do an analysis of the particle s direction, acceleration, motion (speeding up or slowing down), and position.
Rule: 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. Example 6: A particle moves along a vertical coordinate axis so that its position at any time t 0 is given by 1 3 the function s( t) t t 11, where s is measured in centimeters and t is measured in seconds. 6 (a) Find the displacement of the particle during the first 6 seconds. (b) Find the average velocity during the first 6 seconds. (c) Find the expressions for the velocity and acceleration at any time t. v(t) = a(t) = (d) For what value(s), if any, is the particle moving downward? Explain why. Motion Affected by Gravity From our original s( t) 16t v0t s0, we can calculate the velocity function to be vt (), and the acceleration function at (). This is the acceleration due to gravity on earth. FEET s( t) 16t v0t s0 METERS s( t) 4.9t v0t s0 To Find Max Height: To Find When Hits Ground: Set v(t) = 0 Set s(t) = 0 Solve for t Plug that value into s(t)
Example 7: A projectile is launched vertically upward from ground level with an initial velocity of 11 ft/sec. a.) Find the velocity and the speed at t=3 and t=5 seconds. b.) How high will the projectile rise? c.) Find the speed of the projectile when it hits the ground. Example 8: A rock thrown vertically upward from the surface of the moon at a velocity of 4m/sec reaches a height of s( t) 0.8t 4t meters in t seconds. a.) Find the rock s velocity and acceleration as a function of time. b.) How long will it take the rock to reach its highest point? c.) How high did the rock go? d.) How long did it take the rock to reach half its maximum height? e.) How long was the rock aloft? f.) Find the rock s speed when hitting the surface of the moon.
Example 9: A ball is dropped from the top of the Washington Monument which is 555 feet high. a.) How long will it take for the ball to hit the ground? b.) Find the ball s speed at impact. Example 10: Paul has bought a ticket on a special roller coaster at an amusement park which only moves in a 3 straight line. The position s(t) of the car in feet after t seconds is given by s( t) 0.01t 1. t, 0 t 10. a.) Find the velocity and acceleration of the roller coaster after t seconds. b.) When is the roller coaster stopped? c.) When is Paul speeding up? When is Paul slowing down?