- Velocity-Accelerates-Until-0.mp3
- Velocity-Accelerates-Until-0.mp4
- Velocity-Accelerates-Until-I.mp3
- Velocity-Accelerates-Until-I.mp4
- Velocity-Accelerates-Until-intro.mp3
[Intro]
Rollin’ down a hill…
Velocity accelerates until….
[Bridge]
Exponential growth
Exponential velocity
(Indeed)
[Verse 1]
Amass a mass
(Rolling past)
You know…
(Watch ‘er grow)
She’s gonna go
[Bridge]
Rollin’ down a hill…
Velocity accelerates until…
[Chorus]
Rollin’ down a hill
(Faster and faster until)
Rollin’ down a hill
(Bigger, bigger, bigger still)
[Verse 2]
Increase proportional
(to the cube of the radius)
Oh, please! Sensational
(amazing to all of us)
Exponential growth
Exponential velocity
(Indeed)
[Bridge]
Rollin’ down a hill…
Velocity accelerates until…
[Verse]
…until external forces
(friction, resistance, or slope gradient)
…limit the growth… courses…
Reach the limit (that’s it)
[Outro]
Rollin’ (rollin’, rollin’)
Rollin’! (rollin’, rollin’)
A SCIENCE NOTE
As a snowball rolls down a snow-covered hill, its mass and velocity change due to the accumulation of snow and the forces acting on it. Here’s a breakdown of typical changes:
1. Mass Increase:
- Mechanism: The snowball picks up snow from the surface of the hill as it rolls, increasing its mass.
- Rate of Growth:
- The mass increase depends on factors such as the snowball’s surface area, the stickiness and density of the snow, and the snowball’s velocity.
- Snow density can range from 200 to 500 kg/m³, meaning the rate of mass growth varies significantly based on conditions.
- The increase is approximately proportional to the snowball’s surface area, which grows as the square of the radius.
2. Velocity Increase:
- Mechanism: Gravity accelerates the snowball as it moves downhill, increasing its velocity.
- Rate of Acceleration:
- The acceleration depends on the incline of the slope (θ\theta) and frictional forces.
- Friction decreases with steeper slopes or smoother snow surfaces.
Momentum:
- Formula: Momentum is given by p=mv, where m is the mass and v is the velocity.
- Changes:
- As mass (m) increases, momentum increases.
- As velocity (v) increases due to acceleration, momentum increases further.
- Momentum grows at a rate combining both mass accumulation and acceleration, making it nonlinear over time.
3. Typical Observations:
- A small snowball might double in size (diameter) in a short distance on a sticky snow-covered hill.
- Its mass (m) could increase proportional to the cube of its radius.
- Its velocity (v) increases with the slope but may plateau if friction or air resistance becomes significant.
In short, as a snowball gains size, its mass increases significantly, and its velocity accelerates until external forces like friction, air resistance, or slope gradient limit the growth.