Cart rolling down ramp lab with lisa (6th period)
Kinematics:
Kinematics is a type of mechanics that deals with the motion of objects. In this lab, a ramp was elevated by three of the same type of physics textbooks on a table with a flat surface. A weight of 500 grams (0.5 kilograms) was placed on a cart weighing around 99 grams (0.099 kilograms). The cart was then placed at the top of the ramp and was allowed to roll down the ramp until it reached the end of the ramp. At the end of the ramp, a hand stopped the cart from rolling off the ramp. The picture portrayed above shows the cart that was rolled down the ramp in the manner and set-up described above.
The kinematics of the cart can be explained by the position function, Xf=Xi+Vit+(1/2)at^2 where Xf is the final position of the cart, Xi is the initial position of the cart in meters (m), Vi is the initial velocity of the cart in meters per second (m/s), t is the time in seconds (s), and a is the acceleration of the cart in meters per second sqaured (m/s^2). The equation Vf=Vi+at (for final velocity) and Vf^2=Vi^2+2a(Xf-Xi) (for final velocity squared), in which the variables represent the same values as listed previously including Vf representing the final velocity of the cart, can be used to describe the kinematics of the cart.
Based on the data collected, Xi=0 m, Xf= 0.659 m, Vi=0 m/s, Vf=0.877 m/s, t= 1.504 s, and a= 0.583 m/s^2.
By plugging in values based on the data collected for the position function, we get: 0.659 m=0+(0)(1.504)+(1/2)(0.583)(1.504^2) By plugging in values based on the data collected for the final velocity, we get: 0.877 m/s=0+(0.583)(1.504)
By plugging in values based on the data collected for the final velocity squared, we get: 0.768 m/s=(0)^2+2(0.583)(0.659-0)
These equations and their values help us demonstrate and better understand the kinematics of the cart.
Position Vs. Time Graph Velocity Vs. Time Graph (With Line of Best Fit)
The kinematics of the cart can be explained by the position function, Xf=Xi+Vit+(1/2)at^2 where Xf is the final position of the cart, Xi is the initial position of the cart in meters (m), Vi is the initial velocity of the cart in meters per second (m/s), t is the time in seconds (s), and a is the acceleration of the cart in meters per second sqaured (m/s^2). The equation Vf=Vi+at (for final velocity) and Vf^2=Vi^2+2a(Xf-Xi) (for final velocity squared), in which the variables represent the same values as listed previously including Vf representing the final velocity of the cart, can be used to describe the kinematics of the cart.
Based on the data collected, Xi=0 m, Xf= 0.659 m, Vi=0 m/s, Vf=0.877 m/s, t= 1.504 s, and a= 0.583 m/s^2.
By plugging in values based on the data collected for the position function, we get: 0.659 m=0+(0)(1.504)+(1/2)(0.583)(1.504^2) By plugging in values based on the data collected for the final velocity, we get: 0.877 m/s=0+(0.583)(1.504)
By plugging in values based on the data collected for the final velocity squared, we get: 0.768 m/s=(0)^2+2(0.583)(0.659-0)
These equations and their values help us demonstrate and better understand the kinematics of the cart.
Position Vs. Time Graph Velocity Vs. Time Graph (With Line of Best Fit)
Forces:
As the cart rolls down the ramp, there are three forces that act upon it. First off there is the force of gravity that pulls the cart down towards the ground. Then there is the applied force of the ramp that pushes up on the cart. Friction also plays a role in the cart's motion, pushing it backwards.
Free-Body Diagram Newton's Second Law of Motion
Free-Body Diagram Newton's Second Law of Motion
Energy:
Mechanical energy is defined as the total energy possessed by an object while in motion and at rest. There are two types of mechanical energy, kinetic energy and potential energy. Kinetic energy is the energy that an object possesses while in motion. Potential energy is the stored energy of an object based on the object's position. Therefore, mechanical energy is the sum of kinetic and potential energy.
The equation for kinetic energy is KE=(1/2)(m)(v^2) where KE is the kinetic energy in joules (J), m is the mass in kilograms (kg), and v is the velocity in meters per second (m/s). The kinetic energy of the cart increases as the cart goes down the ramp. This is due to an increase of velocity as the cart travels down the ramp.
Based on the data collected, m=0.599 kg (as we're using the mass of the cart and the additional weight on top of the cart), Vi^2= 0 m/s, and Vf^2= 0.768 m/s.
The kinetic energy of the cart at the start of the ramp is: 0 J=(1/2)(0.599)(0)
The kinetic energy of the cart at the end of the ramp is: 0.230 J=(1/2)(0.599)(0.768)
The equation for potential energy is PE=mgh where PE is the potential energy in joules (J), m is the mass in kilograms (kg), g is the gravitational acceleration in meter per second (m/s), and h is the height in meters (m). The gravitational acceleration is always 9.8 m/s^2. The potential energy of the cart decreases as the cart goes down the ramp. This is due to a decrease in height as the cart travels down the ramp.
Based on the data collected, m=0.599 kg, g=9.8 m/s^2, the initial height=0.925 m, and the final height=0.799 m.
The potential energy of the cart at the start of the ramp is: 5.43 J=(0.599)(9.8)(0.925)
The potential energy of the cart at the end of the ramp is: 4.69 J=(0.599)(9.8)(0.799)
Make sure to note that the kinetic and potential energy equations show that not all of the lost potential energy of the cart was converted into kinetic energy.
At the start of the ramp, the total mechanical energy of the cart is KE + PE = 0 J + 5.43 J = 5.43 J.
At the end of the ramp, the total mechanical energy of the cart is KE + PE = 0.230 J + 4.69 J = 4.92 J.
By comparing the total mechanical energy at the end and beginning of the ramp, the different between the total energy at the start and end of the ramp is 5.43 J - 4.92 J = 0.51 J.
Since the matter cannot be destroyed or created, we have to assume that some of the energy was converted to heat through the process of friction.
Energy can neither be created nor destroyed. There are two main energy transformations that occur. The cart starts off at the top of the ramp with potential energy and not kinetic energy. As the cart starts to roll down the ramp, its height from the floor decreases, thus the potential energy decreases. While the potential energy is decreasing this energy becomes converted into kinetic energy. The kinetic energy continues to increase as the cart's acceleration increases.
Kinetic Energy Vs. Time Graph
The equation for kinetic energy is KE=(1/2)(m)(v^2) where KE is the kinetic energy in joules (J), m is the mass in kilograms (kg), and v is the velocity in meters per second (m/s). The kinetic energy of the cart increases as the cart goes down the ramp. This is due to an increase of velocity as the cart travels down the ramp.
Based on the data collected, m=0.599 kg (as we're using the mass of the cart and the additional weight on top of the cart), Vi^2= 0 m/s, and Vf^2= 0.768 m/s.
The kinetic energy of the cart at the start of the ramp is: 0 J=(1/2)(0.599)(0)
The kinetic energy of the cart at the end of the ramp is: 0.230 J=(1/2)(0.599)(0.768)
The equation for potential energy is PE=mgh where PE is the potential energy in joules (J), m is the mass in kilograms (kg), g is the gravitational acceleration in meter per second (m/s), and h is the height in meters (m). The gravitational acceleration is always 9.8 m/s^2. The potential energy of the cart decreases as the cart goes down the ramp. This is due to a decrease in height as the cart travels down the ramp.
Based on the data collected, m=0.599 kg, g=9.8 m/s^2, the initial height=0.925 m, and the final height=0.799 m.
The potential energy of the cart at the start of the ramp is: 5.43 J=(0.599)(9.8)(0.925)
The potential energy of the cart at the end of the ramp is: 4.69 J=(0.599)(9.8)(0.799)
Make sure to note that the kinetic and potential energy equations show that not all of the lost potential energy of the cart was converted into kinetic energy.
At the start of the ramp, the total mechanical energy of the cart is KE + PE = 0 J + 5.43 J = 5.43 J.
At the end of the ramp, the total mechanical energy of the cart is KE + PE = 0.230 J + 4.69 J = 4.92 J.
By comparing the total mechanical energy at the end and beginning of the ramp, the different between the total energy at the start and end of the ramp is 5.43 J - 4.92 J = 0.51 J.
Since the matter cannot be destroyed or created, we have to assume that some of the energy was converted to heat through the process of friction.
Energy can neither be created nor destroyed. There are two main energy transformations that occur. The cart starts off at the top of the ramp with potential energy and not kinetic energy. As the cart starts to roll down the ramp, its height from the floor decreases, thus the potential energy decreases. While the potential energy is decreasing this energy becomes converted into kinetic energy. The kinetic energy continues to increase as the cart's acceleration increases.
Kinetic Energy Vs. Time Graph
The graph of Kinetic Energy VS. Time shows the kinetic energy of the cart over time as it travels down the ramp. The many sharp increases and decreases in the data may be the results of inaccuracies in data collection, but the overall trend of the data shows a general increase of kinetic energy over time. As the velocity increases as time passes by due to an increase in speed of the cart as it travels down the ramp, the kinetic energy increases over time as well.