Finding the Average Acceleration

[Sakyo107]

Homework Statement

"Next, calculate the average acceleration for each change in vector, using the equation..."

Relevant Equations

average acceleration = change in velocity / change in time

Hello, so I am working on a projectile motion lab but I'm not sure what to do right now. Essentially, the lab consisted of my classmates and I using an air table to show that the vertical and horizontal components of projectile motion are independent. During one of our trials, we placed a puck on an incline and pushed it horizontally. The trajectory of the puck was a curve. Now, I found the acceleration between a few points but the issue is that because the puck traveled in a curve, the direction of the acceleration is constantly changing. So I'm not sure how to find the average acceleration.

Please help me ;-; and thanks in advance,

Sakyo

 

[etotheipi]

The horizontal velocity of the puck should remain (fairly) constant, like you said, if the slope is inclined as I think you are describing, there will be vertical (down to the slope) acceleration of around $g\sin{\theta}$. This will result in parabolic motion (i.e. $x \propto t$ and $y \propto t^2$).

It might help if you give more details on the measurements you took. You say you took the acceleration between a few points. What were the "coordinates" of these points and how did you measure the acceleration? Because at the moment I'm struggling to figure out how you recorded the data.

 

[Sakyo107]

Sorry if I didn't give enough details. I did some research and I actually have a different question which would solve the problem and is much easier to explain. If I had the following accelerations:

a1 = 6.0m/s^2 at 60 degrees
a2 = 7.0m/s^2 at 72 degrees
a3 = 7.6m/s^2 at 78 degrees

Would it be correct to find the average acceleration this way?:

(6.0 + 7.0 + 7.6) / 3 = 6.9m/s^2

(60 + 72 + 78) / 3 = 70 degrees

Therefore the average acceleration is 6.9m/s^2 at 70 degrees

 

[etotheipi]

Firstly, the accelerations should only be in the vertical / down the slope direction. How did you calculate those values?

Would it be correct to find the average acceleration this way?:

Just as a note, no. To find the average of vectors, resolve each into components and find the mean of each, then add the component vectors together again. Geometrically, this is the same as drawing all of the vectors from head to tail and then scaling by $\frac{1}{n}$. Then again, I don't see how this applies here.

 

[Sakyo107]

I am not entirely sure to be honest. One of the questions for the lab asks me to find the average acceleration for each trial. If it helps, for the first trial, we let the puck go on the incline (the horizontal component will be 0 i think). The second trial was the same as the first except we pushed the puck horizontally, so the trajectory was a curve. For the last trial, we placed the puck at the bottom of the incline and pushed it diagonally upwards. This created kind of like a parabola.

 

[etotheipi]

I am not entirely sure to be honest. One of the questions for the lab asks me to find the average acceleration for each trial. If it helps, for the first trial, we let the puck go on the incline (the horizontal component will be 0 i think). The second trial was the same as the first except we pushed the puck horizontally, so the trajectory was a curve. For the last trial, we placed the puck at the bottom of the incline and pushed it diagonally upwards. This created kind of like a parabola.

The thing is, there should be no acceleration in the horizontal direction. So I don't know how you have determined an acceleration vector at an angle to some reference. How did you calculate those accelerations?

Are you sure you aren't referring to velocities?

 

[Sakyo107]

Ok so by using an air table, we have this piece of paper with 50 dots on it which recorded the puck's position every 0.02 seconds. This is what our teacher asked us to do: we first choose 10 points on the paper and record the distance from one point to another, so a total of 9 distances. I chose points at an interval of 3 because if I were to choose 10 consecutive points, it would be difficult to measure the distance from one another (since its so small). So, I have selected these 10 points and the difference in time between each is 0.1 second. Now, we divided all 9 distances by 0.1 and got the velocity from point 1 to 2, 2 to 3, and so on. Next, we were instructed to find the change in velocity between each velocity we found, so a total of 8 changes in velocity. Finally, to find the acceleration, we had to divide the change in velocity by the time, 0.1 second. That's how we ended up with 8 accelerations, and now we need to find the average acceleration.

 

[etotheipi]

OK, as far as I can understand you have a set of speeds and the angle of the associated velocity vectors to some reference.

Between any two consecutive velocities, you may write $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$, where $\vec{v}$ is obtained via vector subtraction, if $\Delta t$ is sufficiently small.

I'd suggest using the angles for each velocity to resolve each velocity into a horizontal and vertical component, and then focus on the vertical components for now. The vector equation I wrote then becomes

$a_{y} = \frac{\Delta v_{y}}{\Delta t}$

This should, within the allowances of experimental error, be fairly constant. So I am assuming the question asks you to calculate the mean of these values to find a reliable measure of $a_{y} = g\sin{\theta}$.

Importantly, the derivative of the magnitude of something does not equal the magnitude of the derivative of that thing!

 

[Sakyo107]

That makes so much sense. And yes you're right, we have to then compare it to what we get using ay=gsinθay=gsin⁡θ. Thanks soo much, I'll give it a go and see how it goes!

 

[Lnewqban]

... If it helps, for the first trial, we let the puck go on the incline (the horizontal component will be 0 i think). The second trial was the same as the first except we pushed the puck horizontally, so the trajectory was a curve. For the last trial, we placed the puck at the bottom of the incline and pushed it diagonally upwards. This created kind of like a parabola.

They are asking you to calculate the average scalar value of the vertical acceleration (y-axis of table) for each of the three experiments, which values should be pretty close to each other, as well as constant.
The horizontal acceleration (x-axis of table) for each of the three experiments should be zero or close to it or at least constant (depending on the shooting force).

Having the air table at certain inclination, you guys made the puck follow three different trajectories:

1) A straight line from the higher to the lower edges of the table, which mimics a body in free fall.

2) A semi-parabola (opening to the bottom) from the higher to the lower edges of the table, which mimics the trajectory of a projectile that has been shoot horizontally at certain height.

3) A semi-parabola (opening to the bottom) from the higher to the lower edges of the table, which mimics the trajectory of a projectile that has been shoot from the ground up at certain angle.

For a method of measurements and calculations, please see:
http://eme.eng.ankara.edu.tr/files/2017/02/airtable_booklet-1.pdf

 

[Sakyo107]

Hello, so I am at the very last question on my lab and it states: "Prove that the equation a = g sin x is valid for the magnitude of the acceleration down a frictionless plane inclined at an angle x to the horizontal. I can't seem to understand how I should go about this. A bit of help would be appreciated again ;-;

 

[Lnewqban]

Hello, so I am at the very last question on my lab and it states: "Prove that the equation a = g sin x is valid for the magnitude of the acceleration down a frictionless plane inclined at an angle x to the horizontal. I can't seem to understand how I should go about this. A bit of help would be appreciated again ;-;

It seems that you should compare the measured acceleration for the straight line from the higher to the lower edges of the table, which mimics a body in free fall with the number calculated by that equation.
You know the angle X (in degrees) of the table during the measurements, the value of $sin X$, as well as the value of $g=9.81 m/s^2$

 

[etotheipi]

Yes, for example you could plot $a_y$ against $\sin{x}$ and show that the graph is a straight line of gradient $g$.