STAAR Tutorial: Motion, Speed, Velocity and Acceleration

Transcription

1 Name: Teacher: Period: Date: STAAR Tutorial: Motion, Speed, Velocity and Acceleration TEK 6.8C (Supporting): Calculate average speed using distance and time measurements. TEK 6.8D (Supporting: Measure and graph changes in motion. TEK 8.6B (Supporting): Differentiate between speed, velocity, and acceleration. Motion Distance and Displacement Motion is a change in position of an object, judged by comparison to an assumed fixed frame of reference. For example, if you stand along the edge of a highway, the cars on the highway appear to be moving, as compared to your non-moving position. In reality, no assumed fixed reference point is really fixed or unmoving. The Earth itself and everything on it are constantly moving, when compared to reference points not on Earth. On Earth, we usually judge movement by comparison to some fixed spot on Earth, such as a street intersection, a building or a highway marker. We have a system of latitude and longitude coordinates to describe such a fixed reference point. There are two measurements that we can use to describe the motion of an object: displacement and distance. Displacement is the length between the start and end points, and the compass bearing direction between those points. Distance is the length of the path traveled, including the curves and turns along the way. In the United States, we use feet and miles to measure displacement and distance. In science, and in every other country on Earth, the metric system units of meters and kilometers are used. The illustration below shows both the displacement and distance traveled for the same trip.

2 As you can see for the illustrated trip above, distance and displacement measurements will always be different if the path traveled is not a straight line. In these non-straight-path cases, the displacement will always be less than the distance. Only if the trip is a straight line with no turns will the distance and displacement be the same. Rate of Motion Speed and Velocity Just as there are two ways to measure a change in position (distance and displacement), so too there are two measures of the rate of motion that is, how fast an object is moving: speed and velocity. Speed is the distance traveled divided by the time needed to travel that distance. This is the measurement shown on an automobile speedometer and on the speed limit signs posted along roads. Speed can be measured at a single instant of time (the instantaneous speed ), such as happens when the police measure a car s speed with radar, or you look at a car s speedometer at a single point of time. When the entire distance of a long trip is divided by the time used to complete the trip, the result is average speed. The typical units for speed vary with the length and time of the trip. For short trips, speed can be measured in meters per second (m/s). For longer trips, kilometers per hour (km/hr) would be more appropriate. Average speed can be calculated using the following formula: Distance Average Speed = Sample Problem 1: Karen wants to know how fast she can run. She has a friend time her as she ran a 100 meter sprint at the school track. She completed the sprint in exactly 20.0 seconds. What was her average speed? Answer: 5 m/s. 100 meters = 5 m/s 20 seconds The other method of measuring the rate of motion is velocity, the displacement divided by the time needed to complete that displacement. Because displacement includes a direction as well as a length, velocity at an instant of time changes as the direction changes, even if moving at a constant speed. Sample Problem 2: A family takes a trip to Austin to visit the state capital. The car odometer measures the actual distance traveled as kilometers. The straight-line displacement from the starting point in Carrollton to the end point at the capital in Austin is kilometers. The actual travel time from Carrollton to Austin was 4.0 hours. What was the average speed and velocity for the trip? Answer: Average speed = 87.5 km/hr. Velocity = 78.8 km/hr. (rounded up to nearest tenth)

3 Distance kilometers Average Speed = = = 87.5 km/hr. 4.0 hours Displacement km Velocity = = = 78.8 km/hr. 4.0 hr Acceleration Acceleration is the change in velocity divided by the time over which the change in velocity occurred. The typical units for velocity are meters per second (m/s), and seconds (s) for the time, resulting in an acceleration unit of meters per second squared (m/s 2 ). Acceleration can be calculated using the following formula: (End Velocity Start Velocity) Acceleration = When the starting velocity is slower than the end velocity (that is, when the object is speeding up), the acceleration will be a positive number. When the starting velocity is faster than the end velocity (that is, when the object is slowing down), the acceleration will be a negative number. Sample Problem 3: A car accelerates from a starting velocity of 0 m/s to an end velocity of 10 m/s over a time period of 5 seconds. What is the acceleration? Answer: +2 m/s 2. (10 m/s 0 m/s) 10 m/s = = +2 m/s 2 5 s 5 s Sample Problem 4: A car accelerates from a starting velocity of 8 m/s to an end velocity of 0 m/s over a time period of 4 seconds. What is the acceleration? Answer: -2 m/s 2. (0 m/s 8 m/s) -8 m/s = = -2 m/s 2 4 s 4 s When a free-falling object accelerates due to the pull of gravity, its acceleration is 9.8 meters per second for every second that it free-falls (disregarding the effect of air resistance). For example, after one second of free-fall acceleration, an object will have a speed of 9.8 m/s. After two seconds, its speed will be 19.6 m/s (9.8 m/s 2 x 2s). After three seconds, its speed will be 29.4 m/s (9.8 m/s 2 x 3s). To calculate the free-fall speed of an object (disregarding air resistance), simply multiply 9.8 times the number of seconds of free-fall. The mass of the object does not affect its acceleration.

4 Sample Problem 5: A small metal ball falls off of a building and free-falls for 5 seconds before hitting the ground. What was the ball s speed when it hit the ground, disregarding the effect of air resistance? Answer: 49.0 m/s (9.8 m/s 2 x 5 s). Graphing Speed and Acceleration Speed can be graphed with distance from the starting point plotted on the y-axis and time on the x-axis. The resulting line on the graph represents the speed of the measured object. The steeper the slope, the faster the speed. A horizontal line indicates that the object is not moving (a speed of zero). Acceleration can be graphed with velocity (or speed) plotted on the y-axis and time on the x-axis. The resulting line on the graph represents the acceleration of the measured object. The steeper the slope up to the right, the greater the acceleration (that is, the faster it is speeding up). A horizontal line indicates that the object is moving with a constant speed or velocity (an acceleration of zero). A line that slopes down to the right indicates that the object has a negative acceleration (that is, it is slowing down).

5 Sample Problem 6: Tim takes a trip on his bike to visit two different friends and then return home. He recorded the distance travelled to each friend s house, and the time it took to reach each house and to return home. He also recorded the time he spent visiting at each friend s house. The table of these times and distances is as follows: Home to Friend 1 09 minutes 1.2 km Friend 1 15 minutes 0.0 km Friend 1 to Friend 2 09 minutes 0.9 km Friend 2 21 minutes 0.0 km Friend 2 to Home 06 minutes 0.9 km Draw a graph that correctly represents the trip, including the time spent on each visit. Answer: Practice Problems (Show your work on separate page!) 1. A cyclist leaves home and rides for a distance of 60 km. If the entire round-trip takes 4 hours, what is her average speed? 2. A car travels a distance of 100 km with a displacement of 80 km in 2 hours. What is the velocity of the car? 3. As a roller coaster starts down a hill, its speed is 10 m/s. Three seconds later, its speed is 32 m/s at the bottom of the hill. What is the roller coaster s acceleration? 4. A swimmer slows down from 1.3 m/s to 1.1m/s during the last 20 seconds of the workout. What is the acceleration during this interval? 5. If an object free falls for four seconds, what will its speed be at the end of the four second period, disregarding air resistance? 6. Draw a graph of the following trip, using an appropriate scale and axes: (1) walk 1.2 km in 15 minutes; (2) rest for 10 minutes; (3) jog for 2.0 km in 12 minutes; (4) rest for 15 minutes; (5) walk for 2.4 km in 30 minutes.