Question
The goal of our predictive question is to determine the farthest horizontal distance that a child of given mass (40 kg) can jump off of a swinging swingset. This is an interesting question to explore, as almost anybody who chooses to jump off of a swinging swingset will want to travel the maximum distance possible for maximum fun. So, we will be able to explore that maximum distance for a 40kg child.
Model
We will implement our model by creating a system to execute a sweep of different angles that will give us varying displacements in the x direction to ultimately find the optimal (maximum) distance.
We are neglecting any drag force and frictional force on the seat. In the real world, these forces (particularly air resistance) will play a role when calculating the velocity of the swing, for example, but for the sake of simplicity, we will exclude these external forces. This may result in less accurate results, but we estimate the results to be in the range of reasonable values.
Our parameters include the acceleration due to gravity (9.8 m/s^2), mass of child (40 kg), mass of swing (5 kg, chain mass negligible), length of chain (2.5 m), initial angle (225 degrees), initial angular velocity (0 radians/s, no initial push), and time duration of the pendulum simulation (15 s).
Stage 1: Swing is a pendulum
values | |
---|---|
g | 9.8 meter / second ** 2 |
ChildMass | 40 kilogram |
SwingMass | 5 kilogram |
StringLength | 2.5 meter |
AngleInit | -1.0472 |
t_end | 2 second |
aVelocityInit | 0 / second |
values | |
---|---|
init | angle -1.0472 av 0 / second dtype:... |
Child | 40 kilogram |
r | 2.5 meter |
Seat | 5 kilogram |
g | 9.8 meter / second ** 2 |
t_end | 2 second |
dt | 0.02 second |
angle | av | |
---|---|---|
0.00 | -1.0472 | 0 / second |
0.02 | -1.0465185872800307 dimensionless | 0.06789639165669999 / second |
0.04 | -1.0444819618406629 dimensionless | 0.135739484746347 / second |
0.06 | -1.041089275242371 dimensionless | 0.20347563188653522 / second |
0.08 | -1.0363432057837414 dimensionless | 0.27105052250649103 / second |
... | ... | ... |
1.92 | 0.9675624815849634 dimensionless | -0.7265347787380169 / second |
1.94 | 0.9523861578266541 dimensionless | -0.7907721893986798 / second |
1.96 | 0.9359319099171372 dimensionless | -0.8542905261401317 / second |
1.98 | 0.9182148601370111 dimensionless | -0.9170141735552312 / second |
2.00 | 0.8992516738645995 dimensionless | -0.9788644043599725 / second |
101 rows × 2 columns
Results
Our model generated a plot of the swing angle vs. time as well as angular velocity vs. time. This graph below shows a good confirmation that we have a model that matches up with the physics of a pendulum. This is because you can see in the graph that when the angle is at 0 the angular velocity is at it highest vs. when the angle is at is highest the av is at zero. This matches up well with what pendulums actually do. We put all of these into a sweep series and used this in the second part of the model. We took each angle and its corresponding av and input them into our projectile model to find the maximum horizontal displacement, which is what we will ultimately be recording to answer our question.
Stage 2: Kid is yoted off swing
values | |
---|---|
x | 0 meter |
y | 1 meter |
g | 9.8 meter / second ** 2 |
mass | 40 kilogram |
angle | 45 degree |
velocity | 40.0 meter / second |
t_end | 20 second |
dt | 0.2 second |
r | 2.5 meter |
values | |
---|---|
success | True |
message | A termination event occurred. |
Interpretation
After implementing our sweep, the graph of Range vs. Launch angle shows that the maximum horizontal distance that the child travels after jumping off the swing is between 2.5-3 m, and occurs at an angle of 20 degrees. This makes sense, given that a smaller launch angle will result in a greater horizontal displacement. Also, it seems reasonable to think that a 40kg child will land 3 meters away from the swing, but whether that would be the maximum distance might be up for debate. A limitation of our model lies in the fact that we only accounted for one mass of a child (40 kg), so these particular results would only apply to a child of said mass. In addition, we neglected air resistance and friction caused by the seat of the swing, leading to slight deviations from realistic environments.