Emily Simoskevitz Group Members: Aashvi Shah and Samantha Carella 11/5/18

Uniform Motion Formal Lab

Objective: The objective of the motion lab is to gather experimental data on two constant velocity cars, each with a different constant velocity, and one dynamics cart demonstrating acceleration. Using the data, position vs time graphs will be constructed for all three objects. The formulas learned in the 1-D motion unit will be utilized to derive velocity information from the position vs time graphs. A velocity vs time graph for the dynamics cart will be created from this data.

Theory: This experiment begins with the motion of two constant velocity cars and one dynamics cart accelerating down a ramp. The angle of elevation of the ramp will found by the equation sinθ = (opposite/hypotenuse), where the opposite value is equal to the height above the ground where the beginning of the ramp is and the hypotenuse value is equal to the length of the ramp. The position data will be recorded with a spark timer and ticker tape in 0.1 second intervals. The measurements will be taken by a meter stick, so the units are going to be in centimeters. Each position value has to be converted to meters. This will be done by multiplying the value by the conversion factor: 1m / 100 cm. After all the conversions are calculated, the data points will be graphed to make Position vs Time graphs. A trend line will be fitted for all three objects’ motions. For the two constant velocity cars, the velocity will be derived from information presented by the trend line. The slopes of the trend lines are equal to the velocities of the cars. Since the accelerating dynamics cart’s motion will be represented by a curve, the velocity is constantly changing. To find the acceleration of the cart, average velocity has to be found for each time interval on the Position vs Time graph. The formula, average velocity = slope = △x/△t = (x2 - x1)/(t2 - t1), will be used. The average velocity data will then be graphed on a Velocity vs Time graph. A trend line will be fitted to the points, and the slope will be the dynamics cart’s experimental acceleration. The expected value of acceleration will found using the formula: expected acceleration = 9.8sinθ. The final step will be to calculate the percent error, which will be found by the formula: % error = [(expected value - experimental value) / expected value] x 100.

97/100

Diagram:

Materials: • Constant Velocity Cars (Black and Red) • Dynamics Cart • Ramp/Textbooks (Support for Ramp) • Meter stick • Ticker Tape • Masking Tape • Spark Timer

Procedure: Constant Velocity Cars: 1. Spark timer was plugged in and ticker tape was threaded through the opening. 2. Masking Tape was used to attach the end of the ticker tape to the back of the first constant

velocity car (black). 3. The switches on the car and the timer were activated simultaneously. The car moved forward

with a constant velocity as the ticker tape traveled through the spark timer, which marked a dot after each 0.1 second.

4. After the ticker tape ran out, the car and timer were shut off. The tape was removed from the car, and the meter stick was used to measure the distance between the 0.1 second intervals that the spark timer recorded in dots on the ticker tape.

5. The data for 12 points was written down in table form using centimeters and seconds as the units. The position data then needed to be converted into meters.

6. Steps 1-5 were repeated for the second constant velocity car (red). 7. Position vs Time graphs were constructed for each constant velocity car according to the

data in the the tables, and a trend line was fitted to the points. These graphs were created using the Data Analysis app.

8. The velocities of the cars were derived using the information from the graph.

Dynamics Cart on Ramp: 1. The beginning of the ramp was placed on two textbooks so that the incline remained

constant. 2. The measurements of the height, from the ground to the top of the incline raised by

textbooks, and the length of the ramp were recorded. Then, the angle of elevation of the ramp was found.

3. The spark timer was plugged in, and the ticker tape was threaded through the timer. Masking tape was used to attach the ticker tape to the dynamics cart.

4. The dynamics cart was fitted to the grooves at the top of the ramp. The spark timer was activated and the cart was released from the top of the ramp and proceeded down the surface.

5. The points displayed on the ticker tape were measured and recorded in table form. The points were recorded in 0.5 second intervals instead of 0.1 so that a greater change in distance between points would be visible, and the acceleration would be more accurate. The distance was measured in centimeters, which then had to be converted into meters.

6. A Position vs Time graph for the dynamics cart was constructed using the data. 7. Average velocities for each time interval were found using the slope formula and a Velocity

vs Time graph was created. 8. The acceleration was derived from the Velocity vs Time graph. 9. The expected value of acceleration was found using the value of θ calculated previously.

Then the percent error for the acceleration of the dynamics cart was calculated.

Data:

Analysis:

Position data was measured in centimeters, so conversions from centimeters to meters had to be calculated in order to make the position vs time graphs in the units of meters and seconds.

Conversions:

*The same steps were repeated for all data points.

Black Constant Velocity Car:

t

X

X = 0.1652t - 0.0017143

Black Constant Velocity Car: When t = 0.1s, x = 1.6cm 1.6cm * (1m / 100cm) = 0.016m

Red Constant Velocity Car: When t = 0.1s, x = 1.0cm 1.0cm * (1m / 100cm) = 0.010m

Dynamics Cart When t = 0.5s, x = 1.1cm 1.1cm * (1m / 100cm) = 0.011m

• The equation of the trend line is X = 0.1652t - 0.0017143 using the parameter values presented and the variables of the graph.

• Velocity = slope of the line of motion. For the black constant velocity car, velocity = 0.1652 m/s.

Red Constant Velocity Car:

• The equation of the trend line is X = 0.278t - 0.0239 using the parameter values presented and the variables of the graph.

• Velocity = slope of the line of motion. For the red constant velocity car, velocity = 0.278 m/s.

X = 0.278t - 0.0239

X

t

Dynamics Cart:

• The equation of the trend curve is X = 0.0068731t2 + 0.0208t - 0.00030771 using the parameter values presented and the variables of the graph.

X

t

X = 0.0068731t2 + 0.0208t - 0.00030771

Coordinates for points on the curve:

t (s)X (m)

Finding velocities for the V vs T graph using coordinates:

Average Velocity (Vavg) on a X vs T graph = slope = △x/△t = (x2 - x1)/(t2 - t1) 0s - 0.5s: Vavg = [0.0118m - (-0.000308m)] / (0.5s - 0s) = 0.024216 m/s ~ 0.024 m/s Vavg = 0.024 m/s 0.5s - 1s: Vavg = 0.0312 m/s

1s - 1.5s: Vavg = 0.038

1.5s - 2s: Vavg = 0.045 m/s

2s - 2.5s: Vavg = 0.0518 m/s

2.5s - 3s: Vavg = 0.0584 m/s

3s - 3.5s: Vavg = 0.066 m/s

3.5s - 4s: Vavg = 0.072 m/s

4s - 4.5s: Vavg = 0.08 m/s

4.5s - 5s: Vavg = 0.086 m/s

5s - 5.5s: Vavg = 0.092 m/s

5.5s - 6s: Vavg = 0.1 m/s

• The equation of the trend line is V = 0.0148t + 0.0126 using the parameter values presented and the variables of the graph.

• Acceleration = slope of the V vs T graph. For the dynamics cart, acceleration = 0.0149 m/s2. This is also the experimental value of acceleration.

V = 0.0149t + 0.0126

V

t

Angle of Incline and Percent Error:

The angle of elevation is represented by θ. To solve for θ, the following formula was used: sinθ = (opposite/hypotenuse) where the opposite value is equal to the height above the ground where the beginning of the ramp is (0.1m) and the hypotenuse value is equal to the length of the ramp (1.2m).

Formula with substitution of values: sinθ = (0.1 / 1.2) Take the inverse of both sides: θ = sin-1(0.1 / 1.2) Solve for θ: θ = 4.78° Formula for the expected value of acceleration = 9.8sin θ Expected acceleration = 9.8[sin(4.78)] Expected acceleration = 0.817 m/s2

Experimental acceleration = 0.0149 m/s2

Percent Error = [(expected value - experimental value) / expected value] x 100 Percent Error = [(0.817 - 0.0149) / 0.817] x 100 Percent Error = 98.2%

Conclusion: The goal of this lab was to accurately construct position vs time graphs for two objects with a constant velocity and one object that is accelerating down a ramp. Information derived from the graphs include the trend line equations and the velocities of the two constant velocity cars. Using formulas and the data of the trend curve, a velocity vs time graph could be created for the accelerating dynamics cart. From that graph, the acceleration could be found. To gather the data needed to construct the graphs, ticker tape had to be attached to each object, and a spark timer recorded points at every 0.1 seconds the object had traveled. These distances were measured in centimeters, then converted to meters. Using this data, position vs time graphs could be made. All information derived from the graphs were recorded. Then, average velocities for each time interval of the accelerating dynamics cart were calculated. This data was used to make a velocity

120 cm = 1.2 m10 cm = 0.1 m

θ

vs time graph. From this graph, the acceleration of the motion could be found. The position vs time graph for the black constant velocity car was a line with a positive slope, which is equal to the velocity of the object. The velocity of the black car was 0.1652 m/s. The position vs time graph for the red car was also a line with a positive slope, which is equal to the velocity of the object. The velocity of the red car was 0.278 m/s. These graphs made logical sense because the cars were both moving forward at constant speeds. This is represented by a linear line with a positive slope on an x vs t graph. The position vs time graph for the dynamics cart was a curve. This curve represented motion moving forward speeding up, which is as predicted since the cart increased in speed as it moved forward, traveling down the ramp. The velocity vs time graph was a line with a positive slope since it represents a motion with a constant acceleration forwards speeding up. The acceleration for the dynamics cart was 0.0149 m/s. The percent error was 98%, which is very high. This is due to multiple sources of error. First, the spark timer produced dots on the ticker tape that were scattered and sometimes hard to read. There were oftentimes more than one dot for each 0.1 second interval, and the dots were not in line causing an error in precision of each position. In addition, the data in the velocity vs time graph illustrates that the initial velocity may not have been 0 m/s, which it should have been. This could be due to the fact that the spark timer produced dots that looked “fuzzy” and not legible. The points had to be measured in 0.5 second intervals instead of 0.1 second ones in consequence of this. There were also sources of error for the constant velocity cars. The trend line of the red constant velocity car had more errors than the one of the black car. This could be due to the fact that the car did not reach its full velocity when the switch was initially activated. It may have taken a small increment of time to adjust to the speed. Nonetheless, the objectives of this lab were met and data was accurately found using the data.