More Projectile Motion

What’s the Big Idea‽

So now that we understand the basics of projectile motion, how can we use that knowledge to calculate how a projectile will behave? We can design a setup that allows us to determine the preexisting conditions of a launched projectile, which we can then use to calculate where the projectile will land. After that, further inquiry will allow us to delve into advanced topics dealing with different angles and the location of the projectile-launching mechanism. 

Procedure

Essentially, all we needed to do was design something that would allows us to launch a small metal ball horizontally, with a photogate on the end to measure the velocity of the ball. Instead of just propping the ramp, we used a clamp to attach it to a metal rod. The drama.

The above photograph shows our setup, with the ramp suspended in the air and a photogate at the end (which is attached to a Vernier LabQuest). It was a bit unstable, so we later added a book underneath the bottom of the ramp to keep its height from varying. However, now we needed to worry about how we were going to go about measuring the velocity of the ball as it leaves the ramp. Using a meter stick, we measured the width of the ball (2.3 cm); the photogate allows us to record a single velocity, taking into account how far the ball moves when it interrupts the light (i.e. the ball’s width). However, we noticed that this method of recording the ball’s velocity is prone to error—what happens if the ball blocks the photogate at a height that is to high or to low? Since it’s spherical, the time it takes will appear to be reduced, giving us a faulty velocity. The solution? Lasers!

Above is super awesome laser.

To ensure that the photogate is aligned with the ball, we used a laser to measure the distance of the midpoint of the ball from the top of the table, and adjusted the height of the photogate likewise.

After configuring the LabQuest, all we needed to do was let the ball roll down the ramp, as shown in the video above. To reduce random error, we repeated this process multiple times, recording the velocity each time. To reduce systemic error, we made sure to consistently place the ball at the very top of the ramp.

A screenshot of the LabQuest shows us one of ten trials, our velocities as follows:

• 1.616 m/s
• 1.586 m/s
• 1.603 m/s
• 1.593 m/s
• 1.585 m/s
• 1.619 m/s
• 1.579 m/s
• 1.615 m/s
• 1.624 m/s
• 1.618 m/s

Averaging this data yields a value of 1.604 m/s, our horizontal velocity. We also measured the vertical distance (the height from the ground to the “launch point” of the ball), which was approximately 115 cm. Let’s use our physics knowledge and some kinematics equations to find the horizontal distance the ball travels!

1. First, we’ll need to know how much time it took the ball to fall. We can determine this using the equation Δy = v_iyt+ ½gt².
2. We know what the values of Δy (vertical displacement), v_iy (initial velocity in the y-direction) and g (the force of gravity) are. Substituting them in, we get -1.15 = 0 × t + ½ × -9.8 × t² which simplifies to -1.15 = -4.9t².  Rearranging the terms and simplifying them yields an approximate value of 0.484 s for t.
3. Knowing t, we can now solve for the horizontal distance using the formula v_x = Δx ÷ t. Quickly inputting the values for v_x and t yields a value of about 0.776 m for Δx.

Hooray! This value is very close to the actual horizontal distances we recorded—67 cm, 69 cm, and 71 cm in three independent trials using an identical setup as above. Turns out we can use kinematics to our advantage!

Unfortunately, being in a two-person lab group and running out of time to develop new physics contraptions means that we weren’t able to explore much more in this lab. However, we were able to at least start the next part of the experiment. Above, we fashioned an adjustable ramp using various rods and clamps, allowing to alter the angle of the ball, and thus, its trajectory. We recorded a few trials using a single angle, measuring the ball’s velocity a few times. However, this data has escaped me, and it wouldn’t be of much use anyways seeing as we were unable to finish the portion of the lab.

Takeaways

Though I explained a lot of this above, the first portion of the above allowed us to use the skills we’ve learned previously and apply them to calculate a specific value (namely the horizontal displacement) and then check to see if that value was indeed correct. Which it was! It’s fascinating and beautifully concrete when we can perform and experiment and have the data prove itself.

There were a few questions that this lab left unanswered, unfortunately, mainly because of time restrictions. Though I don’t have the data to back it up, I’ll answer the questions regardless.

• When launching a projectile at an angle, what angle (or angles) produce the greatest range? A 45° angle will produce the greatest range when launching a projectile. This makes sense intuitively, because an angle that is too high will cause the projectile to move a lot in the vertical direction, but ultimately, very little in the horizontal direction. And of course, gravity always gets you down. A range that is too small will not allow the projectile to move far enough because it will hit the ground to early.
• How do the equations developed in the earlier part of the lab need to be modified to deal with projectiles that are not launched horizontally? The equations themselves don’t actually need to be modified—all you need is a bit of trigonometry! By breaking up the initial velocity into horizontal and vertical components, physics continues to work as expected.
• Move your set-up onto the floor. Does this move impact the angle that gives you the maximum range? Though we didn’t do this, it shouldn’t impact the angle that gives us the maximum range. All it does is reduce the amount of time before the ball hits the ground, since the vertical displacement is shortened.
• How does the range of projectiles (across level surfaces) differ for steeper or shallower launch angles? I kind of explained this in the first question, but I can elaborate a bit. As the angle nears 45°, the range of the projectile increases, and likewise, as the angle nears 0° or 90°, the range of the projectile decreases.

The end! For now, at least, for now.

Projectile Motion Video Analysis

What’s the Big Idea‽

Projectile motion occurs when an object simultaneously exhibits both horizontal and vertical motion. Although these motions occur at the same time, they are considered independent. But how can we prove this? In this lab, we use Logger Pro to graph a video of a ball in projectile motion to see how it behaves in the x- and y-directions.

Procedure

The first part of the lab consists of getting our equipment ready and recording an example of projectile motion. To perform the first part of the experiment, we need a meter stick, an iPad, and a ball. First, we lined up the meter stick to the wall to set the scale of the video and prepare it for the video analysis, as shown below.

With that completed, it’s time to film! Projectile motion involves both vertical and horizontal movement, so we throw the ball into the frame of the camera at an angle. Here’s the result:

The second part of the lab is where it gets interesting. After loading the video into Logger Pro, we set the scale of the video, using the measurements provided by the meter stick as a guide. Then, going frame by frame through the video, we plot the center of the ball, as shown below:

Afterwards, we can view the position vs. time graphs. The red dots represent the change in horizontal position and the blue dots represent the change in vertical position.

The shapes of the graphs are the most telling: the change in horizontal position is clearly linear, and the change in vertical position is clearly quadratic. This tells us that no new forces are acting upon the ball in the x-direction and that one consistent force is acting upon the ball in the y-direction, resulting in a gradual change in its velocity.

If we had the velocity vs. time graph, we’d notice that the x-direction would show a completely horizontal line and the y-direction would show a linear graph, indicating a constant acceleration.

Takeaways

So what does all this data mean? What does it show? We know that the original force and velocity can be split up into two separate pieces, one in the horizontal direction and one in the vertical direction. The velocity in the horizontal direction remains constant for the duration of the ball’s flight, however, the velocity in the vertical direction changes, in fact, at a constant rate of -9.8 m/s. Gravity causes the acceleration of the ball downwards, but it only affects it in the y-direction, not in the x-direction as well. These two motions are effectively separate.

However, there are some things that we could have done better in our experiment. Firstly, it would be helpful to record the video a little farther back to see where the ball starts, just for some context. Additionally, having the meter stick in the same plane as the ball while the video is recording (as opposed to measuring the wall) would help eliminate potential sources of error in our graphs. Ultimately, this resulted in strange values for distance in our graphs, none of which made sense. Lastly, we should have saved a version of the graph showing velocity vs. time as well because it would aid in our understanding of projectile motion.