Purpose
The purpose of this lab is to produce graphs of position, velocity, and acceleration vs. time of a cart rolling down an incline.
Theory
I had to find the acceleration and velocity of the cart as it goes down the incline. The calculated values of acceleration and velocity were then put into velocity and acceleration vs. time graphs.
Velocity = Displacement / Time
Acceleration = Change in Velocity / Time
In the first graph, the shape was parabolic, with an equation resembling the third kinematic equation.
0 = (1/2)at^2 + Vit - d
The second graph had a linear shape resembling the first kinematic equation.
Vf = at + Vi
The third graphs also had a linear line that was constant.
A = Constant
Velocity = Displacement / Time
Acceleration = Change in Velocity / Time
In the first graph, the shape was parabolic, with an equation resembling the third kinematic equation.
0 = (1/2)at^2 + Vit - d
The second graph had a linear shape resembling the first kinematic equation.
Vf = at + Vi
The third graphs also had a linear line that was constant.
A = Constant
Experimental Technique
I taped the paper strip to the top of the cart and through the ticker tape timer. I then let it roll down the incline until it hit the bottom and was then able to measure the length between ticks.
Data and Analysis
I noticed that my position vs. time was fairly close to the best fit line at 0.99. My line of best fit for my velocity vs. time graph was also very close at 0.98. The worst line of best fit was my acceleration vs. time graph at 0.005.
I found that at the 75 cm mark, the paper was at the 35th tick mark. This means the velocity was at 4.5 cm/s. Also, at the 15 cm mark, the paper was at the 17.5th tick mark, which means the velocity was at about 2.3 cm/s. The acceleration of the cart is about -.002 cm/s. The acceleration is not quite constant but it is fairly close.
Sample Calculations
Velocity = Displacement / Time
V = 0.4cm - 0.2cm / 1
V = 0.2 cm/s
Acceleration = Change in Velocity / Time
A = 0.3 cm/s - 0.2 cm/s / 1
A = 0.1 cm/s^2
I found that at the 75 cm mark, the paper was at the 35th tick mark. This means the velocity was at 4.5 cm/s. Also, at the 15 cm mark, the paper was at the 17.5th tick mark, which means the velocity was at about 2.3 cm/s. The acceleration of the cart is about -.002 cm/s. The acceleration is not quite constant but it is fairly close.
Sample Calculations
Velocity = Displacement / Time
V = 0.4cm - 0.2cm / 1
V = 0.2 cm/s
Acceleration = Change in Velocity / Time
A = 0.3 cm/s - 0.2 cm/s / 1
A = 0.1 cm/s^2
Conclusion
I set out to create graphs of position, velocity, and acceleration vs. time graphs. I found that the correlation coefficient of the position and velocity vs. time graphs were very strong. In the acceleration vs. time graph, I found that the correlation coefficient was very weak. I was then able to find the velocity at the 75 cm mark and the 15 cm mark. The velocity at 75 cm was 4.5 cm/s while the velocity at 15 cm was 2.3 cm/s. I found that the acceleration is for the most part constant with a few points going off track. This relates to the kinematic equations since you can find initial and final velocity and the distance from interpreting these graphs. Friction may affect the outcome slightly for acceleration and velocity by slightly slowing down the cart. Acceleration has the "noisiest" trend since the data gets less accurate through the more calculations you make.