Uniformly Accelerated Motion

Uniformly Accelerated Motion

Produced by the Physics Staff at Collin College

Copyright © Collin College Physics Department. All Rights Reserved.

Purpose

You will investigate the motion of objects moving with constant acceleration.

Equipment

• Collin Free Fall Simulation (https://www.geogebra.org/m/f3vygzhx)

Introduction

At first glance, it seems obvious from our everyday experience that here on Earth (1) inanimate objects at rest do not move of their own accord, and (2) if you put them into motion they soon return to rest. Based on such observations, Aristotle developed a theory that any object’s natural state is to be at rest and that being in motion is an unnatural (temporary) state. His theory was accepted by the world’s philosophers for over two thousand years.

But Galileo and Newton asked questions about the motion of objects that could not be answered by Aristotle’s theory. In seeking answers to their questions, they ultimately produced a new theory of motion that applied not only to all objects here on Earth, but to everything in the heavens as well. This theory has since been developed more fully and is now accepted by educated people all over the world.

Our current view of motion is that, when all interactions between an object and its environment are considered, any object in constant-velocity motion is just as much in a natural state as is an object at rest. For an isolated object (an object that does not interact with its environment through friction or air drag), no effort or force is required to maintain its constant motion, just as no effort is required to keep it at rest. We can only imagine such an isolated object, of course, because in real life there is always some small friction or drag acting on moving objects on Earth.

The tendency of any object, then, is to remain in whatever state of constant motion it is already in (including constant zero motion – that is, being at rest) as long as no external force is acting on it. This tendency to avoid any change in motion is due to the object’s inertia. Because of an object’s inertia, a net force must be exerted on it to make its state of motion change – from rest to moving, from moving to rest, from moving at one speed to moving at another, or from moving in one direction to moving in another.

The relationships between an object’s position, its speed, its acceleration (rate of change of speed), and time are investigated in a branch of physics called kinematics. The simplest case, the one you will examine in this experiment, is motion in a straight line – called one-dimension motion.

In this experiment, you will:

• Become familiar with some basic measurements of motion.
• Learn to make and read graphs of position, speed, and acceleration vs. time.
• Learn the distinction between average and instantaneous speed and acceleration.
• Investigate one-dimensional motion at constant acceleration.

Dynamics of Motion

For an object in motion, its position is a function of time [x = f (t)]. You can completely describe the motion of any object at any time (in the past, the present, or the future) by describing this function; i.e., by stating the quantitative relationship between its position, speed, and time.

Although a moving object’s position varies with time, either its speed or acceleration may be either variable (can vary with time) or is constant. Furthermore, since any object’s position is relative to some coordinate system, its motion is also relative – what appears as motion to one observer may appear as rest to another observer if the two observers are moving with respect to one another.

Speed (Velocity)

If an object’s position varies with time, we say it is in motion. If the position is constant in time, the object is at rest. We define an object’s displacement as the quantity of change of its position, and its speed as the rate of change of its position. We write the relationship between speed (v), position (x), and time (t) symbolically as

where is the object’s average velocity, Δx is its displacement and Δt is the time interval during which the position changed. Refer to the following figure:

The measurement unit for speed is simply the ratio of distance units to time units. It can be miles/hr, meters/min, feet/sec, furlongs/fortnight, or whatever.

But this simple relationship doesn’t give us all the details of the object’s motion. The most obvious question left unanswered is, “Did the object’s velocity vary during the time interval Δt, or did it remain constant?” If the velocity remained constant, then you know its value was at every instant of time during the interval. But how do you find the value of velocity at any instant if it varied during the time interval? This question leads us to two concepts of velocity.

The first concept is called average velocity. It is the ratio of the object’s displacement to the time interval during which it moved. The above equation expresses average velocity. For example, it’s easy to determine how long a 200-mile trip will take if you know that your average velocity on it will be 50 miles per hour.

On the other hand, the highway police don’t care what your average velocity is over the 200 mile trip. They want to know how fast you are driving at the instant they are observing you (when their radar beam strikes your car), so they can decide whether or not you are exceeding the speed limit at that particular point in time. The highway cop wants to know your instantaneous velocity. Your car’s speedometer indicates your instantaneous speed – your speed at every instant of time. Your trip odometer divided by total trip time is the average speed.

Acceleration

Another question unanswered by these two equations for velocity is, “When the velocity is changing, how fast is it changing?” In other words, what is the rate of change of the velocity? Just as we call the rate of change of position velocity, we call the rate of change of velocity acceleration.

Galileo determined that, in the absence of air drag, all freely-falling objects fall with the same constant acceleration, that is, their velocity increases steadily until they hit the ground, the floor, or whatever. This constant acceleration near the earth’s surface is given its own symbol, g, and its value is g = 9.81 m/s².  This acceleration is always directed downward. If we declare, as is customary, that up is positive and down is negative, then the acceleration of an object in free-fall is: a = – g = – 9.81 m/s².

If the distance fallen by a freely-falling object is y, then the displacement of the object, Δy, is the negative of the distance fallen, and is related to its initial velocity v_o, the time of fall Δt, and of course its acceleration due to gravity g. This relationship is:

where we assume that t_o  = 0. If the object starts its fall from rest (i.e., if v_o = 0), the first term on the right drops out and the distance it falls is simply:

If a moving object’s acceleration varies (not a freely falling object, of course), you can write average and instantaneous values of its acceleration just as you wrote them for velocity. The average value of acceleration over the time interval Δt is:

Similar to velocity, in the limit as Δt → 0 (meaning Δt becomes very small), the average acceleration approaches the instantaneous acceleration at the time t.

Graphing the Motion

The relationships between position, x, time, t, velocity, v, and acceleration, a, can be described analytically, as in the previous equations, or as in the Figure 3.1 (shown previously). A graph of an object’s position vs. time allows you to easily determine its average and instantaneous velocity. You can write the equation defining v as . If x_o = 0, the equation becomes . A graph of x vs. t, such as shown, is a gold mine of information about the object’s motion.

From this graph, we can see that the object’s position when t = 0 is x = 0. Then it starts moving in the minus x direction and reaches x = – 2m when t = 1s. After that, it reverses direction and is back at x = 0 when t = 2s. It continues moving in the positive x direction, reaching the position x = 6m when t = 3s.

By drawing chords between specific points on the curve, we can find the object’s average velocity over a given time interval. In the example shown, the slope of the chord between any two points on the graph is equal to the average value of the velocity during that interval.  For 0 < t < 1s, the average velocity is – 2m/s, and for 1s < t < 3s, the average velocity is + 4m/s.

As the time interval shrinks to a single point on the t axis, the chord becomes the tangent to the graph (the average velocity becomes the instantaneous velocity) at that instant in time.

If the velocity were constant, the slope of the graph would be constant (the graph would be straight). A positive slope indicates a positive value of v (the object’s displacement, x, is increasing with time), a negative slope indicates a negative value of v, and a changing slope indicates a changing value of v.

Similarly, a graph of Δv vs. Δt will be a curve whose slope at any point is equal to the acceleration a. If a is constant, the slope of the graph will be constant (the graph will be straight). A positive slope indicates a positive value of a (the velocity is increasing with time), a negative slope indicates a negative value of a (the velocity is decreasing with time), and a changing slope indicates a changing value of a. The following graphs are for an object thrown straight up then falling back down again.

Figure 3.2 – Four sketches of position and velocity vs. time

Evaluation of Error

If we make a measurement, it is typical to compare that measurement either to other measurements or to an accepted value (or both!). This comparison usually comes in the form of a percent difference (when comparing two measurements) or a percent error (when comparing a measurement to an accepted value).

The percent difference between two measured values x₁ and x₂ is given by:

The percent error between a measured value x_m and an accepted value x_a is given by:

What do these two values really mean? Is a percent error of 5.2% good or bad? Is it the result of having done something wrong, or is it simply the limit of the measuring instrument? In order to make sense of the percent difference/error, we must estimate the measurement error inherent in the experiment. How we estimate measurement error depends on how our value is measured.

Average Deviation

In Part I of this experiment, we will be dropping a ball from a fixed height. Each drop of the ball is called a trial, and we will drop the ball over several trials in this part of the experiment. Because of measurement error, the acceleration of gravity that we calculate will vary from trial to trial — this variation represents an estimate of the measurement error, called an average deviation. Our experimental value for the acceleration of gravity  is just the average of the trial accelerations, but each trial acceleration g is different from this average value. This difference is called the deviation, and is given by . The average

deviation  is then just the average of the trial deviations. How do we use this to interpret error? If the percent error or difference is less than the average deviation, then we can safely say that the error or difference is due to the limits of our measuring devices. However, if the error or difference is greater than the average deviation, something else is going on.

Deviation of Slope

In Part II of this experiment, we will be dropping a ball from multiple heights, and will calculate the acceleration of gravity from the slope of a graph. To estimate the measurement error, we will find what is called the deviation of slope. A straight line with a y-intercept of zero has the equation y = mx, where m is the slope of the line and is given by:

or “rise” over “run”. However, because the y-intercept is zero, each data point on the graph has a “local slope” equal to y/x for that point. If we calculate y/x for each point and identify the maximum and minimum y/x, we can determine the deviation of slope d_s by:

How do we use this to interpret error? If the percent error or difference is less than the deviation of slope, then the error or difference is due to the limits of our measuring devices; if the error or difference is greater than the average deviation, something else is going on.

Systematic Error

If the percent error or difference of a measurement is greater than the measurement error, the “something else going on” is called systematic error. Typically, a systematic error is due to an effect not accounted for in the equations used to calculate the measured value. This effect can be an unexpected physical phenomenon, or it can be inadvertently introduced by the experimental procedure. In questions at the end of the lab, you will be asked to speculate on possible sources of systematic error in both parts of the lab.

Data Collection & Analysis

The virtual version of this lab uses a simulation of the ball drop apparatus pictured on the title page to generate data. That simulation can be accessed here: https://www.geogebra.org/m/f3vygzhx

Below is what you should see when you visit the page. There are brief instructions at the bottom of the simulation page that mirror the procedures below. The controls consist of three buttons and a slider. The slider controls the height that the ball drops from. Two of the buttons allow you to drop the ball and reset the ball to the specified height. The third button toggles between a large ball and a small ball for Part A of the lab. When the ball reaches the pad at the bottom, the time of the fall will be displayed.

Part A. Ball Drop – Single Height, Multiple Sizes

Procedure

• Choose the size of the ball you want to drop first.

• Set the distance from the contact-pad to the bottom of the ball with the slider (any works, 0.5 m is recommended). Use the cm markings in the simulation to record the actual distance to the bottom of the ball. This will be the distance of your drop y. The simulation includes measurement error, so the position probably won’t be exactly what you set it to! However, this height does not change unless you change the slider or the size of the ball.

• Conduct five trials dropping the first ball and record the time t for each trial in Data Table 3.1. This is done by hitting the “Drop Ball!” button, waiting for the ball to hit the pad, and then using the “Reset Ball!” button.
• Using the times recorded on the timer and the distance of your drop y, determine the acceleration due to gravity, g for each trial. Calculate the average of the five trial values of g. Using your average value for g, calculate the percent error of your measurement (Accepted value: g = 9.81 m/s²). Record these results in Data Table 3.1.
• Using the average value of g, calculate the deviation for each trial using the equation: , where is the average value of g. Calculate the average deviation and record your results in Data Table 3.1.
• Repeat steps 1, 2, and 3 with the second ball (of different size) and record your results in Data Table 3.2.

• If requested by your instructor, show example calculations (using your data) of g, deviation, and % error as they request.

Part B. Electronic Timing Ball Drop – Multiple Heights, Single Size

Procedure

• Choose one of the two balls to use for this part of the experiment. Change the height slider so that the drop height is 0.2 m. Record the actual height in your table as y. Drop the ball and record the time t in Data Table 3.3. Calculate and record the value x = ½ t² in Data Table 3.3.
• Repeat step 9 for heights of 0.4 m, 0.6 m, 0.8 m, and 1.0 m.
• Plot a graph of y x (y on the y-axis and x on the x-axis), ideally using a Scatter plot in Excel (you may also do it on a separate sheet of graph paper if your instructor approves). Draw a “best fit” line through your data points and calculate the slope of the graph (you should note that the slope of the graph is equal to the acceleration of gravity). Use the y/x values in your data table to calculate the deviation of slope for the graph. Record your results in Data Table 3.3.
• If requested by your instructor, show example calculations (using your data) of g, deviation, and % error as they request.