POSITION-TIME GRAPHS LEARNING GOALS Students will: Read positions and displacements from position-time graphs. Determine instantaneous and average velocities from position-time graphs. Describe in words motions represented by position-time graphs. DETERMINING TYPES OF MOTION FROM POSITION-TIME GRAPHS Recall that the average velocity of an object can be determined by the slope of a position-time graph. Constant velocity is a straight line on a position-time graph because motion with constant velocity is motion at constant speed and therefore a constant slope. By the end of this unit, you should be able to determine the type of motion of an object based on its position-time graph. Provide examples for each type of motion in the following graphs. Position-Time Graph Characteristics Graph is a horizontal straight line The slope is The velocity is The object is at rest The object is at a constant positive position relative to the reference position The object is stationary at a location to the of the reference position Position (m) [E] Examples Position (m) [N] Position (m) [E] Position (m) [S] Graph is a straight line The slope is The velocity is The object is The object is at a constant position relative to the reference position The object is stationary at a location to the of the reference position Graph is a straight line with a positive slope A non-zero slope represents a constant velocity The object is moving eastward Graph is a straight line with a slope A slope represents a velocity The object is moving Since this graph has a steeper slope we can say it is moving than the object from the previous graph 1
Graph is a straight line with a slope The velocity is also The object is moving westward and travelling from a position east of the reference point. It stops at the Position (m) [E] EXAMPLE 1 MOTION GIVEN BY A POSITION-TIME GRAPH Determine the velocity for the motion shown in the following graph. Describe the characteristics of the object s motion. Position (m [W]) 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 Position vs Time of a Jogger 0.0 0.0 1.0 2.0 3.0 4.0 5.0 DETERMINING INSTANTANEOUS VELOCITY FROM A POSITION-TIME GRAPH If the velocity changes every moment during the motion of an object, no portion of the positiontime graph is a straight line. Remember that each point on the curve represents the position of the object at each instant in time. To determine the velocity of an object at any instant, physicists use tangents. A tangent is a straight line that touches a curve at only one point and is parallel to the curve at that point. Each tangent has a unique slope that represents the instantaneous velocity at that instant in time. 2
EXAMPLE 2 INSTANTANEOUS VELOCITY ON AN POSITION-TIME GRAPH The position-time data for an all-terrain vehicle (ATV) approaching a river are shown on the following Graph. Find the instantaneous velocities at 2.0 s, 3.0 s, and 5.0 s. Position (m [E]) 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 Position vs Time 0.0 1.0 2.0 3.0 4.0 5.0 Tangent Line Position vs Time 40.0 30.0 Rise (Δ) 20.0 10.0 Run (Δ) 0.0 0.0 1.0 2.0 3.0 4.0 5.0 Position (m [E]) Position (m [E]) Position (m [E]) 40.0 Position vs Time 30.0 Rise (Δ) 20.0 Run (Δ) 10.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 40.0 Position vs Time 30.0 20.0 10.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 A tangent line at 2.0 s is drawn and only touches the point at 2.0 s. The slope of the tangent line is the instantaneous velocity at 2.0 s. slope Δ Δ At 3.0 s, slope Δ Δ slope slope At 5.0 s, the tangent is a horizontal line. slope Δ Δ slope 3
EXAMPLE 3 POSITION-TIME GRAPH FOR TWO PEOPLE Two friends start walking on a football field in the same direction. Person A walks twice as fast as person B. However, person B has a head start of 20.0 m. If person A walks at 3.0 m/s, find the distance between the two fiends after walking for 20.0 s and determine who is ahead at this time. Sketch a position-time graph for both people. Position-Time Graph Position (m) CLASS WORK 1. At the end of the school day, Lois and Clark say goodbye and head in opposite directions, walking at constant rates. Lois heads west to the bus stop while Clark walks east to his house. After 3.0 min, Lois 300 m [E] and Clark is 450 m [W]. Graph the position of each student on one graph after 3.0 min. 2. Two rollerbladers, Peter and Parker, are having a race. Peter gives Parker a head start of 5.0 s. If Parker travels 100.0 m [right] in 20.0 s and Peter travels 112.5 m[right] in 15.0 s, a. Graph the motions of both rollerbladers on the same graph. b. Find the time, position and displacement when Peter catches Parker. 4
PRACTICE POSITION-TIME GRAPHS The position - time graph for an object in motion is shown below. Position (m [N]) 10 8 6 4 2 0-2 Position-Time Graph A B G C 0 1 2 3 4 5 6 7 8 9 F H I -4-6 D E a) Write a short story that could represent the motion of this graph. b) What is the displacement in the intervals: i) AD ii) EI c) What is the distance traveled in the intervals: i) AE ii) DH d) What is the velocity in the intervals: i) BC ii) FG e) What segments of the trip have a zero velocity? Explain? f) What is the average speed in the intervals: i) AE ii) DH iii) AH g) What is the average velocity in the intervals: i) AC ii) DH 5
FINDING VELOCITY FROM A CURVED POSITION-TIME GRAPH Position (m [up]) 7 6 5 4 3 2 1 0 Position vs. Time for a Ball Thrown Upward 0 0.5 1 1.5 2 2.5 1. The ball is on its way up from 0s until and it returns to the ground at. On the way up, its speed is and you can tell because the slope is getting. On the way down, the speed is and you can tell because the slope is getting. At the maximum height, its speed is. 2. Find the average velocity of the ball from 0s until 0.60 s. 3. Find the average velocity of the ball from 1.40 s until 2.2 s. 4. Find the instantaneous velocity of the ball at a. 0.10 s b. 1.60 s 6
ACCELERATION LEARNING GOALS Students will Define acceleration as the rate of change in velocity. Understand that acceleration is a vector quantity. THE RELATIONSHIP BETWEEN THE DIRECTIONS OF VELOCITY AND ACCELERATION The same signs for velocity and acceleration mean the object is speeding up. The opposite signs for velocity and acceleration means the object is slowing down. CHANGE IN VELOCITY PER UNIT TIME Velocity is a vector quantity. The average velocity is defined as the displacement per unit time. Another way of saying the same thing is that velocity is the rate of change of position. Acceleration,, is the change in velocity per unit time or the rate of change of velocity. A. From the definition, develop an equation using words and variables to find the average acceleration for an object experiencing a change in velocity. The direction and magnitude of vector, depends on the magnitude and direction of both vectors and. The units for acceleration are m/s divided by s which is m/s/s or m/s 2. B. Provide numerical examples of final and initial velocities for an object undergoing acceleration if, The magnitude of its velocity changes, while its direction remains the same; The direction of its velocity changes, while its magnitude remains the same; There is a change in magnitude and direction of its velocity; EXAMPLE 1: ACCELERATION DUE TO BRAKING A man is driving his car at a velocity of 100 km/h (needs conversion to m/s) on a straight road. He sees a traffic jam at a distance and slows down to 12.5 m/s in 3.0 s. What is the average acceleration of the car? Provide a FULL solution. 7
UNIFORM AND NON-UNIFORM MOTION Draw displacement vectors between the plane s initial and final positions at each time interval. (Each of the plane images is separated by the same interval of time.) C. Which image above represents uniform motion and non-uniform motion? Based on the displacement vectors that you drew, how do you know and what is the difference? SPEEDING UP AND SLOWING DOWN When an object travels with increasing velocities, we say that it is speeding up. Conversely if it travels with decreasing velocities, we say that it is slowing down. Both velocity and acceleration are vector quantities. D. In the following two diagrams, indicate where the velocity is increasing and decreasing as well as the signs (positive or negative) of the velocity and acceleration. Speeding up occurs when the signs of the velocity and the acceleration are the same and slowing down occurs when the signs are opposite, regardless of the direction of motion. 8
EXAMPLE 2: ACCELERATION IN THE OPPOSITE DIRECTION An ATV is undergoing non-uniform motion. Describe the motion with respect to the displacement, velocity and acceleration of the ATV as it moves to the right. Assume right is positive. EXAMPLE 4: SPEEDING AWAY FROM A STOP SIGN A car at a stop sign accelerates from rest at a uniform 1.0 m/s 2. How fast is it going after 6.5 s? (in m/s and km/h) EXAMPLE 5: CHANGING DIRECTION A ball is rolled up a hill with an initial velocity of 12 m/s [uphill]. After 7.2 s have passed, the ball is rolling downhill at 1.4 m/s. What was the ball s acceleration? 9
SPECIAL CASE: CONSTANT ACCELERATION You learned earlier that the ONLY way to properly calculate the average velocity of an object is to use total displacement over total time. You also did a problem where you demonstrated to yourself that the formula + 2 gives the wrong answer. However, there are some cases where this equation will work and give the correct answer. If an object is undergoing a constant acceleration, then its average velocity can be found using the above equation. DO NOT USE THIS IF YOUR ACCELERATION IS NOT CONSTANT THROUGHOUT THE ENTIRE TIME INTERVAL. The above formula is actually the formula to calculate the mid-point between the initial and final velocities. This mid-point is the average if the acceleration is a constant. This has implications for finding instantaneous velocities on position-time graphs. If you have a position-time graph showing constant acceleration and you want the instantaneous velocity at say 2.0 s, then you can draw a tangent at 2.0 s. HOWEVER, you can also choose a time interval where 2.0 s is the mid-point (like 1.0 s to 3.0 s or 0.0 s to 4.0 s) and calculate the average velocity by finding the slope of the line joining the ends of that time interval. REMEMBER, THIS ONLY WORKS IF THE POSITION-TIME GRAPH IS FOR CONSTANT ACCELERATION. A graph that shows this is a parabola you will learn more about these shapes in math class. Notice how both slopes in the above graph are equal. This graph represents constant acceleration. The slope at the mid-point of 3.0 s is equal to the slope from 1.0 s to 5.0 s. 10
PRACTICE PROBLEMS 1. A catapult accelerates a rock from rest to a velocity of 15.0 m/s [S] over a time interval of 12.5 s. What is the rock s average acceleration? 2. As a car approaches a highway on-ramp, it increases its velocity from 17 m/s [N] to 25 m/s [N] over 12 s. What is the car s average acceleration? 3. A squash ball with an initial velocity of 25 m/s [W] is hit by a squash racket, changing its velocity to 29 m/s [E] in 0.25 s. What is the squash ball s average acceleration? 4. How long does it take a radio-controlled car to accelerate from 3.2 m/s [W] to 5.8 m/s [W] if it experiences an average acceleration of 1.23 m/s 2 [W]? 5. A speedboat experiences an average acceleration of 2.4 m/s 2 [W]. If the boat accelerates for 6.2 s and has a final velocity of 17 m/s [W], what was the initial velocity of the speedboat? 6. What is the average acceleration of a sports car that increases its velocity from 2.0 m/s [W] to 4.5 m/s [W] in 1.9 s? 7. If a child on a bicycle can accelerate at an average rate of 0.53 m/s 2, how long would it take to increase the bicycle s velocity from 0.68 m/s [N] to 0.89 m/s [N]? 8. a) While approaching a red light, a student driver begins to apply the brakes. If the car s brakes can cause an average acceleration of 2.90 m/s 2 [S] and it takes 5.72 s for the car to come to rest, what was the car s initial velocity? b) What is the significance of the direction of the initial velocity and that of the acceleration? 9. What is the average acceleration of a tennis ball that has an initial velocity of 6.0 m/s [E] and a final velocity of 7.3 m/s [W], if it is in contact with a tennis racket for 0.094 s? Answers: 1. 1.20 m/s 2 [S] 2. 0.67 m/s 2 [N] 3. 2.2 x 10 2 m/s 2 [E] 4. 2.1 s 5. 2.1 m/s [W] 6. 1.3 m/s 2 [W] 7. 0.40 s 8. a) 16.6 m/s 9. 140 m/s 2 11