代写PHYS110-001 Position, Velocity and Acceleration Lab Report代做回归

Position, Velocity and Acceleration

Lab Report

PHYS110-001

Due Date: 8:00 AM on 08/26/2024

Abstract

We studied the position, velocity and acceleration carts on level and inclined tracks using carbon tape timers.  We found that the cart moved with a nearly constant speed of 0.48 m/s on a level track.  On the inclined track, the cart accelerated with an acceleration of 0.34 m/s2.  As the track was inclined, the component of the acceleration due to gravity along the track increased causing the cart to accelerate.

Introduction

In this lab, we are measuring the position, velocity and acceleration of a cart moving along a track. In the first portion of the lab, this is done with a level track where we expect that the velocity of the cart to be constant.  In the second part of the lab, we used an inclined track so that the cart accelerates.  We can measure the acceleration of this cart by observing its change in position as a function of time.  The position, velocity and acceleration of an object can all be related using the kinematic equations of motion as long as the acceleration is constant.  In this lab, we will be using these two equations to calculate the velocity and acceleration of the cart after measuring its position as a function of time.

Procedure

We wanted to measure the velocity and acceleration of carts using tape timers. The tape timers work by creating a carbon dot on a piece of paper at well-defined intervals of time.  In this lab, we used the 10 Hz setting so that the timers made 10 marks per second on the paper.  In the first part of the lab, we used a level track for the cart.  The schematic of the setup is shown in Figure 1 below.  The cart was started on the left hand side of the track.  The tape timer was turned on and the cart was given a gentle push so that it moved to the right hand side.  Once the cart reached the right hand side, the tape timer was turned off.  We examined the tape to make sure that the timer created a series of dots.  To find the position and velocity of the cart we measured the positions and separations ofthe dots on the tape.

Figure 1. Schematic for the level track experiment showing cart, tape timer, track and paper tape.

For the second part of the lab, we used a slightly inclined track.  The schematic of the setup for this portion of the lab is shown in Figure 2 below.  The left hand side of the track was inclined using a 4 cm high block of wood.  The tape timer was turned on and the cart was released from rest. When the cart reached the bottom of the track, the tape timer was turned off. Once again, we examined the tape to make sure that the timer created a series of dots on the tape.  To find the position, velocity and acceleration of the cart, we measured the dots on the tape as discussed below.

Figure 2. Schematic for the inclined track experiment showing cart, tape timer, track, paper tape and block.

Theory

In this lab, we measured the separation between dots created by the tape timers.  We needed to convert these separations into the velocity and acceleration of the cart as a function of time.  The tape timers create dots at constant intervals of time.  Therefore, we can determine the average velocity v of the cart between two dots as

where Δxn is the separation between the nth and (n-1)th dot and Δtn  is the change in time between those dots.  Similarly, we can determine the average acceleration a of the cart

where Δvn  is the change in velocity between the nth and n-1 dot and Δtn  is the change in time between those dots.

We can also determine the velocity and acceleration of the cart using the kinematic equations.

The position of the cart as a function of time is given by

where x0  is the initial position, xf  is the final position and v0  is the initial velocity.  If we start the cart from the origin (x0  = 0)  and on a level track with no acceleration  (a = 0), this equation simplifies to

So, we can obtain the velocity of the cart from the slope of the graph of position as a function of time.

When the cart is on an incline, it will accelerate.  However, we start the cart from rest (v0   = 0), so equation (3) becomes

Then from the slope of the graph of position versus time squared, we obtain the acceleration of the cart.  We can also obtain the acceleration of the cart using the equation for the velocity of the cart as a function of time,

So, if the cart starts from rest v0 = 0, we can simplify equation (6) to read,

Therefore, the slope of a plot of the velocity as a function of time gives the acceleration of the cart.

Sample Calculation and Results

We have measured the position of the cart as a function of time for both the level track and the inclined track.  The raw data is shown in table #1 for the level track and table #2 for the inclined track.  These tables are attached at the end of the report.  As a dot was created every 0.1 seconds, we have calculated the time by multiplying the dot number by 0.1 s.  We have measured the position of the dots from the first point to calculate the displacement of the cart.

For the level track data, we have plotted the position of the cart as a function of time in graph #1. The slope of the best fit line to this plot gave us the average velocity of the cart and it was 0.479 ± 0.004 m/s.  We have calculated the average velocity of the cart between each pair of dots using equation (1). We have determined the velocity between the first pair of points as follows

The velocities at the other locations were calculated in a similar manner.  In graph #2, we plotted this calculated velocity of the cart as a function of time.  We see that the cart slows done slightly as a function of time.  From our data the average velocity was 0.48 ± 0.03 m/s.

For the inclined track, we once again plotted the position of the cart as a function of time. This is shown in graph #3.  This graph is parabolic showing that the cart is accelerating as it moves down the track.  We have re-plotted this position data as a function of time squared in graph #4. This graph is now linear and we can fit a straight line to the data to give the average acceleration of the cart.  We find that the best fit line has a slope of 0.188 ± 0.001 m/s2.  The acceleration of the cart is twice this slope, so we find an acceleration of 0.376 ± 0.002 m/s2.  We have calculated the velocity of the cart as a function of time using equation (1) in the same manner as the level track.  The results are plotted in graph #5.  We find that the velocity of the cart linearly increases with time.  The slope of the best fit line gives the acceleration of the cart.  We find the slope is 0.346 ± 0.002 m/s2.  Lastly, we calculated the acceleration of the cart from the change in velocity using equation (2).  We have determined the acceleration at the second point as follows

The accelerations at the other locations were calculated in a similar manner. The results are plotted in graph #6.  We see that the acceleration is nearly constant as a function of time but it is slightly smaller near the end of the data.  We found an average acceleration of 0.34 ± 0.07 m/s2.

Graph #1: This graph plots the position of the cart as a function of time for the level track. The points are equally spaced in position indicating that the cart move with a constant velocity.  The red line indicates a best-fit line to the data.

Graph #2: This graph plots the average velocity of the cart as a function of time for the level track.  The velocity slows down the longer the cart travels.

Graph #3: This graph plots the position of the cart as a function of time for the inclined track.  The points become farther spaced in position as a function of time indicating that the cart is moving with an increasing velocity.

Graph #4: This graph plots the position of the cart as a function of time squared for the inclined track.  The points lie on a straight line indicating that the cart moves with a constant acceleration.  The red line indicates a best-fit line to the data.

Graph #5: This graph plots the average velocity of the cart as a function of time for the inclined track.  The points are equally spaced in velocity indicating that the cart moved with a constant acceleration.  The red line indicates a best-fit line to the data.

Graph #6: This graph plots the average acceleration of the cart as a function of time for the inclined track.  The average acceleration values are nearly constant.