Lab Report 2 1f1u4z

Lab Report 2: Introduction to Kinematics Using the Motion Sensor

PHYS 261-002 Author: H. Patton Lab Partners: K. Santos, C. Evans Date: 1/7/14

Objective: The objective of this lab was to provide an introduction to the concept of kinematics through the use of a motion sensor. The position, velocity, and acceleration were introduced. They were calculated using the Logger Pro software. The data shows a clear relationship between position, velocity, and acceleration.

Theory: A motion sensor is a device that uses sound beams to estimate the position of an object by how far the sound travels. Since position is found, we then have velocity and acceleration. There are several equations to represent these kinematic . Position is represented as the symbol x. The velocity of an object can be represented in of time t as: vt  

x t

Eq. 1

where Δx is the change in position on the interval t to t+Δt. Velocity is also known as the derivative of the position. This means that it is the slope of the tangent line of the position graph at any given time t. Velocity can be negative or positive depending on the direction the object is moving towards the motion detector. When moving towards the detector, the velocity is positive and the opposite when moving away. The relationship between acceleration and velocity is very similar to that of velocity and position. Acceleration can be represented in of time t as:

at  

v t

Eq. 2

where Δv follows the same interval as the stated above for the position. The acceleration function is the derivative of the velocity function. Since acceleration is the slope of the tangent line of the velocity graph, if the velocity remains constant, the acceleration is zero. In this case, there is a special formula to calculate the position. It is as follows:

x  vt  x0

Eq. 3

This equation shows a linear position change and a constant velocity. Next, there is another case where the acceleration is nonzero. The formulas for velocity and position are as follows:

v  at  v0

Eq. 4

1 2 Eq. 5 at  v0t  x0 2 These equations show that position is a quadratic and velocity is linear. In the equations, x0 is the initial position at time t=0 and v0 is the initial velocity at time t=0. x

Procedure: To set up the experiment, a motion sensor was attached to the Vernier Lab Pro interface which was then connected to a computer with access to the Logger Pro software. For each procedure, there are 40 data samples taken per second for 15 First, the motion sensor is used to measure position, velocity, and acceleration of a person walking towards and then away from the sensor. The results are then plotted in Logger Pro. Next, the motion sensor was used to measure position, velocity, and acceleration of a cart moving on a level frictionless track. The results were then potted in Logger Pro. The data was then analyzed for relationships between position, velocity, and acceleration. Lastly, the motion sensor was used to measure position, velocity, and acceleration of a cart moving on an inclined track. The track was inclined using a wooden block. There were two procedures that took place with the inclined track. One where the cart was let go from the top and another where someone pushed the car from the bottom to the top and let it roll back down to the bottom.

Data: The first procedure where a person walked toward the motion sensor is shown on Figure 1 in blue while the procedure where a person walked away is in orange.

Figure 1: The three graphs above represent the position, velocity, and acceleration with respect to time for the first procedure

The second procedure was where a cart was pushed along a frictionless level track. This procedure is shown below in Figure 2.

Figure 2: The three graphs above represent the position, velocity, and acceleration with respect to time for the second procedure.

The third procedure is the one where a cart is let go from the top of an inclined track. This procedure is shown below in Figure 3.

Figure 3: The three graphs above represent position, velocity, and acceleration with respect to time for the second procedure.

Lastly, the fourth procedure is the one in which the cart is pushed from the bottom of the inclined track and allowed to roll up and back down. This procedure is shown below in Figure 4.

Figure 4: The three graphs above represent the position, velocity, and acceleration with respect to time for the fourth procedure.

Analysis: For the first procedure, the position graph is linear. This means that the velocity graph is constant and the acceleration graph is zero. Figure 1 backs up this assumption. The position graph was constant at the beginning because the person has not started moving before the motion sensor had started measuring. Velocity initially starts at zero and remains constant as the person moves towards or away from the sensor. For this procedure, the acceleration is constant at zero. For the second procedure, the slope of the linear fit of the position graph should have been equal to the average velocity, and the slope of the linear fit of the velocity graph should have been equal to the average acceleration. These values are shown below in Table 1.

Uncertainty Average Value Slope Position -0.221 m .0053 m/s Velocity -0.2205m/s 0.04996m/s .071 m/s^2 Acceleration 0.04384m/s^2

Table 1: The values of the second procedure. The slope of the position should equal the average velocity and the same with velocity and acceleration.

The values mentioned above are correct based on the graphs. While there is a value for acceleration, the value is very close to zero so it is safe to assume that the acceleration is essentially zero. For the third procedure, the position graph should have made a curve. As you can see from Figure 3, the graph is not much of a curve. That can be blamed on the error of the motion sensor, which seemed to not give an accurate reading sometimes. Though, the velocity graph is linear and the acceleration graph is constant. The comparison of the slope of the velocity graph and the average acceleration is shown below in Table 2.

Velocity

Average Value Slope .1747 m/s

Acceleration 0.06393m/s^2

Uncertainty .0423 m/s .9464 m/s^2

Table 2: The values of the third procedure. The slope of the acceleration should equal the average acceleration.

The slope of the velocity and the average acceleration are not exactly equal but they are close so this error can be attributed to the motion sensor. The acceleration is very minimal is this procedure. The expected acceleration was .333 m/s^2 based on the formula a=g*(h/b) where g is acceleration due to gravity, h is the height of the block, and b is the base of the track. This number is not extremely far off the measured value, but it is not as close as it could be. For the fourth procedure, the position graph made a parabolic curve. This means the velocity graph was linear and the acceleration graph is constant. The maximum point on the position curve is where the slope of the tangent line is zero. This means this is where the velocity is equal to zero when the cart stops moving at the top of the track. Below are the average acceleration and the slope of the velocity. Uncertainty Average Value Velocity Acceleration

-0.2588m/s^2

Slope -0.2557 m/s

Table 3: The values of the fourth procedure. The slope of the velocity should be roughly equal to the average acceleration.

0.058 m/s^2

The average acceleration never changes signs because the slope of the velocity is constant, thus keeping the acceleration constant. Even though the sign of the velocity changes, the slope never does so that makes the acceleration constant and the same sign throughout. Conclusion: Overall, this experiment accomplished its goals. A greater knowledge of kinematics was obtained throughout this experiment. Each procedure had its errors, but overall it turned out well. Some explanations for the various errors are that the motion sensor could have malfunctioned. Another explanation could be that the motion sensor was picking up the

movements of the lab participants rather than the cart at some times. Both of these explanations are very logical reasons as to why these errors may have occurred. A way to fix this next time would be to test this experiment with several motion sensors to get an accurate reading of the cart. Overall, this experiment was successful.