Kinematics of the System
Kinematics is in common words is the study of motion and the science behind it. This section will observe the motion of the the cart rolling down the ramp and show graphs to complement the research.
A. Text Description
The cart is initially held by the physics teacher's hand at the top of a ramp (The 5 book ramp in particular). As the physics teacher let go of the cart, cart began to accelerate down the ramp with a relatively smooth ride down except for the occasional squeak causing a few bumps in the road.
B. Using graphs and formulas
Using the equations depicting the kinematics of the system, we could further our understanding of what is going on in the cart rolling down the ramp scene. There are three formulas which describe the kinematics and they are each explained below.
Xf =Xi + vit + 1/2at^2
The formula above could be used to solve acceleration, since tracker is able to give us everything but acceleration. We substitute the values we have for final distance, initial distance, time, and initial velocity. The steps below show how our group found what the acceleration was. The position graph's measurements were off by a power of ten(because of computer failure), therefore we must divide the x-final by 10. 775.45/10 = 77.545.
77.545 = 0.00 + 0(1.2) + (1/2)a(1.2^2)
77.545 = (1.44)a/2
a = 107.701 cm/s^2
a = 1.07701 m/s^2
77.545 = (1.44)a/2
a = 107.701 cm/s^2
a = 1.07701 m/s^2
We also have the two velocity equations which give help us figure out the final velocity now finding the value of the acceleration
Vf = Vi + at
Vf = 0 + 1.2(107.701)
Vf = 129.2412 cm/s , rougly what the velocity is in the first formula.
Vf^2 = Vi^2 +2a(Xf-Xi)
Vf^2 = 0 + 2(107.701)(77.545)
Vf = 129.2414 cm/s
Vf = 1.292414 m/s
somewhat close in value to the others but vary because as tracker makes a few errors here and there such as with velocity, squaring it and finding combining values leads to more varied results.
Vf = 0 + 1.2(107.701)
Vf = 129.2412 cm/s , rougly what the velocity is in the first formula.
Vf^2 = Vi^2 +2a(Xf-Xi)
Vf^2 = 0 + 2(107.701)(77.545)
Vf = 129.2414 cm/s
Vf = 1.292414 m/s
somewhat close in value to the others but vary because as tracker makes a few errors here and there such as with velocity, squaring it and finding combining values leads to more varied results.
The graphs below complement our findings with the formulas and show us where we obtained the values used to find acceleration. The leftmost one shows how as time increases, the position from the initial point increases at an exponential rate. Because this is so, we can deduce from logical thinking and the formulas above that the cart is accelerating and isn't travelling at constant velocity. The second graph also shows that this statement is true by showing us how Vx (velocity) is increasing at a robustly constant rate.
Forces of the System
Forces act on everything in the universe and in this section we will observe the forces that cause the cart rolling down the ramp to move the way it does. There are two forces in particular which have the most impact on the way the cart acts. Gravity is one of them which exerts a downward force on the cart forcing it to go to the center of the earth. However, since the strong nuclear force holds the ramp as well as the Earth together, the cart won't sink through the wood as well the Earth.
While every other force such as weak nuclear force, and electromagnetic stay balanced, gravity and the strong force remained unbalanced shown with the evidence that the cart is accelerating. Below shows a free body diagram which models what forces are acting on it as it rolls down the ramp. As we see below, the force applied or the momentum which causes the cart to accelerate is going rightward, and the friction is going leftward. Since the gravity and strong force is unbalanced, gravity is causing the cart to be pulled down the ramp.
While every other force such as weak nuclear force, and electromagnetic stay balanced, gravity and the strong force remained unbalanced shown with the evidence that the cart is accelerating. Below shows a free body diagram which models what forces are acting on it as it rolls down the ramp. As we see below, the force applied or the momentum which causes the cart to accelerate is going rightward, and the friction is going leftward. Since the gravity and strong force is unbalanced, gravity is causing the cart to be pulled down the ramp.
Using Newton's Second Law, we can see how much force is being exerted on the cart, substituting values we found from the previous equations.
F = ma
F = 996g(1.07701m/s^2)
F = 1072.70
F = 1.07 N (units were made proper i.e. grams to Kg.)
F = 996g(1.07701m/s^2)
F = 1072.70
F = 1.07 N (units were made proper i.e. grams to Kg.)
Energy and its application in the system
There are two kinds of energy which are kinetic energy and potential energy. Studying the cart and its motion will include studying how these two kinds of energy are taking place. Below shows the kinetic and potential energy shown mathematically. Kinetic energy basically shows the energy a motion or force has. The highest amount of kinetic energy the cart would have would be the velocity of when the cart is going the fastest, or towards the bottom of the ramp before it was stopped. The formula below shows the kinetic energy just before the cart departed the ramp.
The highest amount of kinetic energy the cart would have would be the velocity of when the cart is going the fastest, or towards the bottom of the ramp before it was stopped. The formula above shows the kinetic energy just before the cart departed the ramp. The graph to above and to the right shows the kinetic energy over the period of time it was going down the ramp. The graph is increasing at a curve because the cart is accelerating and has a few bumps due to errors in tracker or the fact that motion can never completely be smooth without some friction.
PE = mgh
PE = (0.996)(9.8)(0.23)
PE = 2.244984 J
PE = (0.996)(9.8)(0.23)
PE = 2.244984 J
After calculating all of this, it becomes necessary to state that the energy being shown has not been gained or lost, but merely transferred. If every transaction of energy were to be added up, the amount would be exactly the same due to the Law of Conservation.