Lab 5: Trajectories

We are starting to explore the more calculus intensive aspect of multidimensional kinetics. This experiment serves as sort of a review. Essentially, we will a let a ball roll down a ramp to hit some point on the floor, and then attempt to use math to find where the ball trajectory crosses certain points.

Involved in this experiment is a ring stand, which together with a clamp serves as a stand to raise an aluminum v-channel to an incline. Two wooden boards with a channel in them holds the second v-channel stably. A steel ball is placed near the top and allowed to roll down to hit the floor, where a carbon paper is placed.

The carbon paper will be marked upon impact, keeping track of where the ball ends up. We'll do it 5 times just to make sure it ends up reasonably in the same place, otherwise note its uncertainty.

Looking closely, there is a mark in the center of that carbon paper.

The goal now is to calculate v₀, the speed just as the steel ball leaves the ramp. To do this, we need to find the height of the table, and the distance of the landing point from the edge of the table. Since it's difficult to get a ruler perfectly perpendicular to the floor, we use a self-fashioned plumb bob, a length of string tied to a weight that will utilize gravity to make sure it goes straight down. We then measure the length of string. And from where the plumb bob meets the ground out to the carbon paper. In this manner, we found the height h to be 94 cm, and distance out from the table d to be 75.5 cm.

One way of solving the velocity is to first find the time it takes for the steel ball to reach the ground, and since horizontal and vertical components are independent, we only need to consider the vertical in account of gravity. We assume that the v-channel we laid on the table is perfectly horizontal, thus leaving the ball with no initial vertical velocity. So we set the equation up as such:

 h = 0.5 a t²

0.94 m = 0.5 (9.8 m/s²) t²

t² = (9.4 m) / (4.9 m/s²)

t² = 0.1918 s²

t = 0.328 s

Now, we plug this figure in the horizontal components to find velocity, such that:

v₀ = d / t

v₀ = (0.755 m) / (0.438 s)

v₀ = 1.7328 m/s

We will now place a wooden plank over the carbon paper where the ball lands, such that it forms an angle α with the floor, leaned up against the table, and figure out the distance d down this ramp that the ball will travel.

If we know how to calculate the trajectory of projectiles, then it should be possible to predict the path theoretically, such that our prediction should correlate with experimental results, give or take a margin of error. We could express this error as uncertainty. We first note that the point where the ball lands on the ramp must satisfy two conditions: It must be within the trajectory of the ball, and it must intercept the plank. We express the x and y coordinates in terms of the relevant variables:

Let's use the x-component to solve for time t, then plug it into the y-component to solve for d:

d cosα = v₀ t

t = (d cosα) / v₀

d sinα = 0.5 a t²

d sinα = 0.5 a (d cosα / v₀)²

d sinα = 0.5 a d² cos²α / v₀²

sinα = (d² / d) 0.5 a cos²α / v₀²

v₀² sinα = d 0.5 a cos²α

d = ( v₀² sinα ) / ( 0.5 a cos²α )

Plugging the values in, we get:

d = [ (1.7328 m/s)² sin47.4º ] / [ 0.5 (9.8 m/s²) cos²47.4º ]
d = [ (3.0026 m²/s²) 0.7361 ] / [ (4.9 m/s²) 0.4582 ]
d = (2.2102 m²/s²) / (2.2450 m/s²)
d = 0.9845 m

 Since we haven't learned how to propagate uncertainty yet, we can round this to a reasonable 0.98 m.

Okay. We are ready to set up the plank. We do one run with the steel ball, then tape carbon paper in its vicinity in order to mark the exact landing point, as we've done before. Then, we measure the distance and obtain our experimental value to see how close it is to our theoretical value.

No, those aren't termite holes. They are the marks left by the carbon paper. Measured from where the top of the plank meets the v-channel, we get (1.01 ± 0.02) m. This is fairly close to our theoretical value. Our variance is 1.01 m - 0.98 m = 0.03 m. We can divide it by our theoretical value to get a simple relative uncertainty, such that 0.03 m / 0.98 m = 3.1%.

There are conceivably many places where this error could have been introduced. Even with the plumb bob, our process involved holding a meter stick against a length of string, or the plank. This is admittedly inexact. The plank had also barely reached the v-channel, requiring props to hold it in place. This might have caused small shifts after the ball hit it for the first time. It is also possible that the cell phone app used to measure the angle was inexact, either inherently or because of the contours of the phone or the plank. The analog angle would have required even more guesswork. Our calculations also assume that the table and floor are flat (and not slanted), whereas this is not necessarily the case. Nevertheless, that our theoretical and experimental values are within 3.1% of each other can be considered a relative success, and shows some basis to the formulas.