SPEED & ACCELERATION

PART I: A DISTANCE-TIME STUDY OF SPEED Speed is composed of two fundamental concepts, namely, distance and time. In this part of the experiment you will take measurements of time and distance to calculate the speed of a car moving in a constant speed. For an object traveling at a constant speed, 퐝퐢퐬퐭퐚퐧퐜퐞 퐒퐩퐞퐞퐝 = 퐭퐢퐦퐞 Turn on the motor of the car. Set the car down at the starting line and start the stopwatch. Determine the time it takes for the car to travel a distance of 2.00 meters. Repeat for three trials and determine the average time.

Average Time Trial 1 Trial 2 Trial 3 (seconds)

5.25

 Calculate the average speed of the car. (3 significant digits) speed = 2.00 m / 5.25 s = 0.381 m/s

 From your results, how many seconds would it take the car to travel 1.50 meters? (3 significant digits) time = distance / speed = 1.50 m / 0.381 m/s = 3.94 s

 From your results, how many meters would the car travel in 1.50 seconds? (3 significant digits) Distance = speed x time = 0.381 m/s x 1.50 s = 0.572 m

 If the car were required to travel twice the distance, or 4.00 meters, how would this change the time required? It would take twice time.

Is the relationship between speed and distance directly proportional or inverseley proportional?

 If the speed of the car were doubled, how would this change the time required for the car to travel 2.00 meters? It would take half the time.

Is the relationship between speed and time directly proportional or inverseley proportional?

PART II: ACCELERATING OBJECTS Measurement of the motion of falling objects is difficult because the speed increases rapidly, approximately 10 m/s every second it is falling. The distance the object falls becomes very large, very quickly. Galileo slowed down the motion by using inclined planes. The less steep the incline, the smaller the acceleration. Around the year 1600, Galileo claimed that a falling object will gain equal amounts of speed in equal times. He reasoned that as the object fell: 1) it increases in speed the longer (or farther) it fell.

2) the increase in speed each second was the same. In other words, the acceleration of a falling object is constant.

3) the distance an object moved is proportional to the square of the time it is falling. In other words, the ratio of d / t2 was constant for a given angle of inclination of the ramp.

Speed & Acceleration 1 PROCEDURE: STEP 1: An inclined plane has been set up which measures 2.400 meters in length which has been divided into 6 equal lengths with yellow tape that marks the positions where you will release a marble down the ramp. A block is placed at the end of the ramp to catch the ball. Roll the ball down the ramp at each position and record the time it takes for 3 separate trials.

Table 1: Time Data of an Accelerating Object Distance Time (seconds) (meters) Trial 1 Trial 2 Trial 3 Average 0.400 1.81

0.800 2.50 1.200 3.10

1.600 3.58

2.000 3.90 2.400 4.25

 Plot this data on the distance vs time graph provided on the final page of the laboratory.

The average speed of the ball along the ramp can be 퐝퐢퐬퐭퐚퐧퐜퐞 퐀퐯퐞퐫퐚퐠퐞 퐬퐩퐞퐞퐝 = determined with the formula: 퐭퐢퐦퐞

The final speed (instantaneous speed) of the ball at the 퐅퐢퐧퐚퐥 퐬퐩퐞퐞퐝 = ퟐ 퐱 퐀퐯퐞퐫퐚퐠퐞 퐬퐩퐞퐞퐝 end of the ramp can be dermined using the formula:

퐜퐡퐚퐧퐠퐞 퐢퐧 퐬퐩퐞퐞퐝 The acceleration of the ball down the ramp can be 퐀퐜퐜퐞퐥퐞퐫퐚퐭퐢퐨퐧 = determined using the formula: 퐜퐡퐚퐧퐠퐞 퐢퐧 퐭퐢퐦퐞

 Using the final speeds, determine the change in speeds between points (assume the first point is 0 m and 0 s). Using the change in speed and change in time determine the acceleration of the ball at each distance.

Data from Table 1: d vs t data of an accelerating object Distance Time Average Final Speed Change in Change in Acceleration (meters) (seconds) Speed (m/s) Speed (m/s) Time (s) (m/s2) (m/s) 0.400 1.81 0.221 0.442 0.442 1.81 0.244

0.800 2.50 0.320 0.640 0.198 0.69 0.287

1.200 3.10 0.387 0.774 0.134 0.60 0.223

1.600 3.58 0.447 0.894 0.120 0.48 0.250

2.000 3.90 0.513 1.026 0.132 0.48 0.275

2.400 4.25 0.565 1.130 0.104 0.35 0.297

NOTE: 1) FINAL SPEEDS INCREASE WITH DISTANCE. 2) CHANGE IN TIME DECREASES WITH DISTANCE. 3) ACCELERATIONS ARE RELATIVELY CONSTANT.

Speed & Acceleration 2 STEP 2: On the same inclined plane, orange tape marks have been placed at increading lengths moving up the ramp. Roll the ball down the ramp at each position and record the time it takes for 3 separate trials.

Distance Table 2: Time data Time Between Intervals (meters) Trial 1 Trial 2 Trial 3 Average (seconds) 0.150 t1 = 1.10 s ------

0.600 t2 = 2.20 s t2 – t1 1.10

1.350 t3 = 3.00 s t3 – t2 1.20

2.400 t4 = 4.00 s t4 – t3 1.00

 Calculate the average times for the 3 trials and the time between intervals.

In this experiment the distances between starting points was not constant.

 How many times longer is the second starting point than the first? 4 times

 How many times longer is the third starting point than the first? 9 times

 How many times longer is the fourth starting point than the first? 16 times

 Examine the times between intervals. How do they compare? They are similar (relatively).

Speed & Acceleration 3 PART III: PROJECTILE MOTION Gravity acts in the vertical direction. For an object that is falling from a height, the object’s speed increases by 9.80 m/s every second it is falling. For an object that is moving upwards, the object’s speed decreases by 9.80 m/s2. Notice that while the object is falling, the distance the object falls is greater each second it falls. On the way upward, the distance decreases since the speed is also decreasing. The changes in motion of an object moving upward is the same for an object that is moving downward since gravity is acting in the same direction – accelerating the object back to earth.

When an object is thrown through the air horizontally, the speed of the object is relatively constant. Thus the object Horizontal component covers the same distance each second. Projectile motion is a combination of the vertical component which is affected by gravity and the horizontal component which is the speed of the Vertical object. component Projectile To summarize: motion Horizontal motion distance = speed x time 풅 = 풔 풕 풅 speed = distance ÷ time 풔 = 풕 Vertical motion ퟏ distance = ½ x acceleration due to gravity x time2 풅 = 품 풕ퟐ ퟐ

velocity = acceleration due to gravity x time 풗 = 품 풕

The goal of this portion of the laboratory is to predict where a ball will land on the floor when released from an inclined plane. The final test of your measurements and calculations will be to position an empty can so that the ball lands in the can the first time. height

range PROCEDURE: Step 1: The horizontal component.

 Assemble your ramp so that the bottom of the ramp is about 10 cm from the edge of the table.

Measure the length of ramp: 1.220 meters

 Determine the average time required to roll down the ramp for 3 separate trials:

Trial 1: ______(s) Trial 2: ______(s) Trial 3: ______(s) Average 1.50 s

Speed & Acceleration 4 The average speed of the ball along the ramp can be determined using the formula: 퐝퐢퐬퐭퐚퐧퐜퐞 퐀퐯퐞퐫퐚퐠퐞 퐬퐩퐞퐞퐝 = 퐭퐢퐦퐞

The final speed, also called the instantaneous speed, is the speed of the ball at the end of the ramp. This can be dermined using the formula: 퐅퐢퐧퐚퐥 퐬퐩퐞퐞퐝 = ퟐ 퐱 퐀퐯퐞퐫퐚퐠퐞 퐬퐩퐞퐞퐝

 Calculate the final speed of the ball at the end of the ramp:

Ave. Speed = 1.220 m / 1.50 s = 0.813 m/s

Final Speed = 0.813 m/s x 2 = 1.63 m/s

Step 2: The vertical component. Height of table: 0.800 m

 Next, determine the time the ball is in the air when falling from the height of the table till it hits the floor. Use 1 equation 푑 = 푔 푡2, rearrange to solve for time: 2 2 푑 푡 = √ 푔

ퟐ (ퟎ. ퟖퟎퟎ풎) t = √ 풎 = 0.404 s ퟗ. ퟖퟎ 풔ퟐ

This the the time it takes for the ball to fall from the table to the floor.

 Using the final speed of the ball at the end of the ramp, determine the horizontal distance (the range) the ball falls through the air.

Distance = speed x time = 1.63 m/s x 0.404 s = 0.659 m

This is the distance the cup is to be placed from the edge of the table.

Step 3: The Test

Now that you’ve determined the range of the ball, place a piece of tape on the floor tance from the edge of the table and place the can at that spot. Place the ramp about 10 cm from the edge of the table and align the ramp with the can so that when the ball is released it will fall into the can.

Release the ball!

Speed & Acceleration 5

Distance - Time Data of an Accelerating Object 2.400

2.000

1.600

Distance 1.200

0.800

0.400

0.000 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

Time  Examine the trend in distance – time data plotted on the graph. Does it appear to be a straight line or a curve? What does the shape of the graph imply about the slope? Slope is increasing. What does the slope represent? Speed – the speed of the ball down the inclined plane is increasing.

Final Speed - Time Data of an Accelerating Object 1.50 1.40 1.30 1.20 1.10 F 1.00 i 0.90 n 0.80 a 0.70 l 0.60 0.50 S 0.40 p 0.30 e 0.20 e 0.10 d 0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Time

 Examine the trend in the instantaneous speed – time data plotted on the graph. Does it appear to be a straight line or a curve? What does the shape of the graph imply about the slope? Constant. What does the slope represent? Acceleration. The acceleration of the ball moving down the inclined plane is constant.

Speed & Acceleration 6