Cart Rolling Down a Ramp: Three Books
Kinematics
3 Equations:
xf = xi+vit+1/2at²
We can use the velocity vs. time graph and the position vs. time graph to calculate the acceleration of the cart. We can find any of the variables as long as we have the other ones. Xf is final displacement and Xi is initial displacement, t is time and a is acceleration. According to the graphs, the initial displacement is 0 and the final displacement is o.95 meters. We also the know the initial velocity is 0 and that the time it took for the cart to travel down the ramp was 1.2 seconds. When we substitute all of the information in the equation our result is:
0.95 meters = 0+ (0)(1.2 seconds) + 1/2 (a) (1.2)²
0.95 = 0.72 (a)
0.95/0.72= a
1.32 = a
Acceleration = 1.32 meters per second²
vf= vi + at
Looking at this equation, the initial velocity (Vi), acceleration (a), and time (t) are variables that already have measurements. We can combine this information and find the final velocity (Vf). The initial velocity is 0, the acceleration is 1.32 meter per second², and the time is 1.2 seconds.
Vf = 0 + (1.32) (1.2)
Vf = 1.58 meters per second
Vf² = Vi² + 2aΔx
We can use this final equation to determine if all of our calculations so far have been accurate. We know the values of all the variables in this equation. Final velocity is 1.58 meters per second, initial velocity is 0, acceleration is 1.32 meters per second², and the change in displacement is 0.95 meters.
1.58²= 0 + 2(1.32) (0.95)
2.50 =2.51
Due to rounding the numbers, the quantities won't be exactly equal, but since the difference between the two values is one-hundredth, it proves that all of the values calculated above are accurate.
3 Equations:
xf = xi+vit+1/2at²
We can use the velocity vs. time graph and the position vs. time graph to calculate the acceleration of the cart. We can find any of the variables as long as we have the other ones. Xf is final displacement and Xi is initial displacement, t is time and a is acceleration. According to the graphs, the initial displacement is 0 and the final displacement is o.95 meters. We also the know the initial velocity is 0 and that the time it took for the cart to travel down the ramp was 1.2 seconds. When we substitute all of the information in the equation our result is:
0.95 meters = 0+ (0)(1.2 seconds) + 1/2 (a) (1.2)²
0.95 = 0.72 (a)
0.95/0.72= a
1.32 = a
Acceleration = 1.32 meters per second²
vf= vi + at
Looking at this equation, the initial velocity (Vi), acceleration (a), and time (t) are variables that already have measurements. We can combine this information and find the final velocity (Vf). The initial velocity is 0, the acceleration is 1.32 meter per second², and the time is 1.2 seconds.
Vf = 0 + (1.32) (1.2)
Vf = 1.58 meters per second
Vf² = Vi² + 2aΔx
We can use this final equation to determine if all of our calculations so far have been accurate. We know the values of all the variables in this equation. Final velocity is 1.58 meters per second, initial velocity is 0, acceleration is 1.32 meters per second², and the change in displacement is 0.95 meters.
1.58²= 0 + 2(1.32) (0.95)
2.50 =2.51
Due to rounding the numbers, the quantities won't be exactly equal, but since the difference between the two values is one-hundredth, it proves that all of the values calculated above are accurate.
![Picture](/uploads/3/8/7/7/38773679/7568808.jpg?414)
Velocity vs. Time:
Overall, the velocity vs. time graph is fairly linear because as time increases velocity is also increasing. The upward slope of the graph proves that acceleration is increasing as the time passes,
but towards the end it stabilizes which is when the cart rolls of the ramp. Increases in acceleration occur because when the cart travels down the ramp, the slope becomes steeper and steeper causing the acceleration to increase. The velocity vs. time graph isn't perfectly linear because data is never perfect.
Overall, the velocity vs. time graph is fairly linear because as time increases velocity is also increasing. The upward slope of the graph proves that acceleration is increasing as the time passes,
but towards the end it stabilizes which is when the cart rolls of the ramp. Increases in acceleration occur because when the cart travels down the ramp, the slope becomes steeper and steeper causing the acceleration to increase. The velocity vs. time graph isn't perfectly linear because data is never perfect.
Forces
Newton's Second Law: The second law states that an object is dependent on the net force being applied to the object and the mass of the object, (F=Mass x Acceleration). This pertains to the cart rolling down the ramp, because when the cart goes down the ramp, the acceleration also increases. The mass is always constant through the duration of the experiment, but the acceleration is increasing, hence increasing the amount of force applied.
F= Mass x Acceleration
F= 0.099675 kg x 1.32 m/second²
F= 0.131571 Newtons
![Picture](/uploads/3/8/7/7/38773679/4544759.jpg?427)
1) F1 is equal to the force of gravity.
2) F2 is equal to the normal force.
3) F3 is equal to the force of friction.
Photo Credits:
http://ffden-2.phys.uaf.edu/211_fall2004.web.dir/Jeff_Levison/Freebody%20diagram.htm
Energy
![Picture](/uploads/3/8/7/7/38773679/2655968.jpg?446)
Kinetic Energy Vs. Time: The kinetic energy in this
graph increases, because the as time increases, the cart continues rolling down the ramp. This means that kinetic energy, which is the energy of motion, is increasing because the cart is speeding up.
The Law of Conservation of Energy:
The Law of Conservation of Energy states that energy can neither be created or destroyed, but can only be transformed. An example of energy being transformed was when potential energy changed to kinetic energy. At the the top of the ramp, the cart had the most potential energy and as the cart started rolling down it started to transform into kinetic energy. In a closed system, the total amount of energy is constant.
The Kinetic Energy:
The equation for kinetic energy is 1/2mv².
KE= 1/2 * 0.099675 kg * 1.58 m/s²
KE= 0.07874325 Joules
Potential Energy:
The equation for potential energy is mass x gravity x height.
PE= 0.099675 kg x 9.8 m/s² x 0.13 m
PE= 0.12698595 Joules