Understanding the Force Acting on a Toy Train During Deceleration

good morning today our primary objective will be to determine the force

acting on a toy train while slowing down to a stop and so this is what the diagram looks

like we have a force that's opposing the motion and the question is how large is that

force we'll only be interested in the force acting while the train is actually

slowing down or stopping secondary objective is to determine if the force changes

when mass is added to the train if we make the train heavier does that have any effect on the force

and so to accomplish this secondary objective we're going to have the train at three

different weights you can see the top weight there 473.7 grams

then we're going to add something to the train 626.1 grams and finally we're gonna add two

objects to the train 829.6 grams is the stopping force affected by the mass

and so the question is how are we going to measure force well we could use the formula f net

equals m a however today we're going to use another concept

we're going to use the concept of network equals the change in kinetic energy

so ultimately in a few moments what you're going to see is that general equation network equals the

change in kinetic energy can be rewritten as force times distance equals the initial kinetic energy

and so to see this derivation the question is what is network to answer that question we'll start off

with what is work work is a measure of energy transfer gain or loss of energy

when a force acts on an object over a distance or displacement the definition of work is force times

distance times cosine theta now this definition assumes that the

force acting is constant so throughout our investigation today we will assume that the force that's

slowing down the train is constant and so what is network network is the sum of the individual

work done by each force so if there's two forces we see a sum of two different work

values if there's a third force then that equation will be adjusted to write plus w

force three well today in our investigation we'll assume that while the train is slowing

down there's only one force acting we'll also assume that that one force is

constant and so continuing on with our derivation network

equals the change in kinetic energy since there's only one force acting we only have to be concerned

with one work value the work done by the force acting while the train is slowing down

so the equation once again given for work is force times distance times cosine theta

this is the definition of the force it's opposing the motion and this is the distance d

d is the distance from when the train begins to slow down to when it comes to a complete

stop so what is theta theta for the situation is 180 degrees

why 180 degrees well if the two vectors the force and displacement

are pointing in the same direction theta is zero degrees when the force vector is perpendicular

to the displacement vector theta is 90 degrees and finally in this situation as you can

see the force vectors in one direction the displacement vectors in the opposite direction

when they're in opposite directions theta is 180 degrees and so in this situation you can see

force and displacement vector are in opposite directions and so substituting

we know that cosine of 180 is negative one now we're going to simplify

and there's our equation and so the question is what is delta e k

well any change in a value in science is the difference between the final state and the initial state

in this case the change in kinetic energy is the difference between the final kinetic energy and the initial

kinetic energy well we know the final kinetic energy is zero

this is because we know the train comes to a complete stop the final speed is zero

therefore we have this statement here remember that kinetic energy is half mv squared where m is the mass

and v is the speed and so we end up with this and further simplifying we end up with

this final equation force times distance equals half mass times speed squared

so to measure force we need these three quantities distance initial speed and mass

so to determine the distance and initial speed we are going to use the motion sensor that is circled in the

diagram there how this motion sensor works is that it emits

sound you can see the sound represented by the white arrow the sound is going to encounter a

surface the flat surface that i've put at the front of the train when sound encounters that flat surface

it's going to reflect it's going to reflect in the same direction from where it came from

reflected sound is detected by the sensor so the same sensor that creates the sound

also can detect sound the computer of the sensor can determine the time it takes

for the sound to travel from when it left the sensor to when it came back

the computer also knows the speed of sound so those two variables the time it takes for the wave to travel back and

forth and the speed of sound the computer can determine the distance

between the front of the train and the sensor it can also determine the speed of the

train in real time sensor via bluetooth sends data

on time speed and distance to an app this app is usually on some sort of mobile device

or a tablet and so this is a screenshot of the information that is displayed

in an app as you can see velocity on the left side and position on the right side

for all trials a screen recorder of the data is captured mainly while the train is slowing down

and so we have an acceleration phase here and during the acceleration phase there

is no motion information this is because the sensor there which is circled in white

emits sound in that direction and it has nothing to reflect off of for the sensor to work sound has to

reflect back to the sensor and so right around at this position

once again the sensor is circled in white at around that position the sensor will begin to detect

the presence of the train and can measure its speed and position today i'd like you to copy down the

following table and this is what you'll be submitting to me eventually

all right for the first five trials the mass of the train is shown there here we go

so this is what the trial looks like i did that five different times here's what the data looks like

it's very quick because the train stops relatively quickly so from the video you just saw of the

screen capture of the data i have extracted two still images the first image where it says initial

speed is the speed of the train just before i turn the train off so the speed of the train before i

turned the train off was 0.41 meters per second the position the exact position at which

i turned the train off happened to be 1.03 meters once i've turned the train off it then

goes to a stop it came to a complete stop at a final position of 1.12

meters and so with the initial position of 1.03 meters

this is the position of the train as i turned it off and the final position of 1.12 meters

we're going to take the difference of those two positions and that will be our distance

and so there's our distance for trial a our speed is circled that's the initial speed

that's the top speed the train achieved just before i turned it off so in this situation

for trial a our speed is 0.41 meters per second and so here's our formula force times

distance equals half mass times speed squared the mass is listed there

so please substitute your numbers into this equation to get the force remember if you leave

the mass in grams force will be the unit of milli newtons so if you want your force

in newtons you need to convert your mass into kilograms once again if you want your force

in newtons you need to convert your mass into kilograms please be aware of that

all right here's trial b and here's the data for trial b trial c

here's the data for trial c trial d here's the data for trial d

and trial e the data for trial e next we add the small little plane on

top of the train to increase its mass so this is the mass for the next five trials

f g h i n j and there's our train with now the added

mass and here's one sample run all right here's the screen capture data

for all the trials together here's trial f g

h i nj

in addition to the data being displayed on the app you can export the data to an excel file and that's what i've

done here so i've on purpose chosen the specific data

where the top speed happened so at 6.9 seconds that's when we had the top speed of the train for trial

f and you can see it there it's 0.47 meters per second at 6.9 seconds i turned off the train and then it

coasted it coasted until it came to a complete stop at 7.6 seconds

and you can see at 7.6 seconds the speed is zero here's the data for g

h i and j

finally we have trials k l m n and o and i have two ships attached to the train to increase the mass

by an even greater amount here is the data for k

l m n

and oh once again just like the previous five trials i've exported the data from the

sensor into an excel file at 10.7 seconds that was the point at which i shut off the train

and at that point it had a speed of 0.37 meters per second and its position was 1.33 meters

and by 11.3 seconds the train came to a stop traveling about 11 centimeters in total

the data for l m n and o

now while slowing down we made this assumption we assumed the force acting was constant

is this assumption valid does the data support this assumption

so let's see let's look at our data this is one of the reasons why for 10 of the trials i gave you the data in

greater detail if the force is constant then the

acceleration should be constant given that the mass does not change

so is the acceleration constant does the data support this well if the acceleration is constant

then the change in speed should be constant so now the question is is the change in speed constant

while the train is slowing down in this situation from six and a half to seven and a half seconds

well the answer is no as you can see from 6.5 to 6.6 seconds the train speed has

decreased by 0.07 meters per second and once again during the next time interval from 6.6

to 6.7 seconds it again drops by 0.07 meters per second the speed of the train

so so far so good it looks like it's constant acceleration however notice during the next time

interval from 6.7 to 6.8 seconds the drop in speed is 0.08 meters per

second and then it only decreases its speed by 0.05 meters per second

and during the next time interval the decrease in speed is 0.10 meters per second

and these are all the decreases in speed during the time intervals of one tenth of a second

while the train is slowing down so for the most part it seems steady at 0.07 meters per second

however there are certain times where that decrease changes and so

the acceleration is fairly constant but it's not absolutely constant therefore the force that was calculated

during this experiment is at best an approximation as the force is not constant since the

acceleration is not constant now if you have this train set you'll know

that the front of the train looks like this notice it has some curves

and yet as you can see i replaced the front of the train with this flat surface

and so the question is why the flat surface we have the flat surface to ensure the

sound reflects back to the sensor without the reflected sound

the distance and the speed cannot be measured and so with a curved surface you may

have a situation where the incoming sound hits the curved surface but then it

reflects at an odd angle and it does not reflect back to the sensor if this situation

happened then the sensor's data would basically return values that don't make any

sense and so i've been working with the sensor long enough to know that to avoid this complication i just made a

flat surface and for the most part in fact for every data point i saw

all the data points made sense meaning that the sound was reflecting properly off of the front surface

so i hope you enjoyed today's activity i hope it gave you some insight into another way of determining forces

but also some insight into the force acting on this train