Connecting Position, Velocity, Acceleration of Functions, Integrals, Straight Line Motion - Calculus

Position, velocity, and acceleration are linked by calculus: velocity is the derivative of position (rate of change), and acceleration is the derivative of velocity (rate of change of velocity), or the second derivative of position. To find the related function, you can differentiate to go forward (position to velocity to acceleration) or integrate to go backward (acceleration to velocity to position).

💡Derivatives (Going Forward)
• Position (\(s(t)\)) to Velocity (\(v(t)\)): Velocity is the instantaneous rate of change of position with respect to time. To find the velocity function, you differentiate the position function.
\(v(t)=\frac{d}{dt}s(t)\)
• Velocity (\(v(t)\)) to Acceleration (\(a(t)\)): Acceleration is the instantaneous rate of change of velocity with respect to time. To find the acceleration function, you differentiate the velocity function.
\(a(t)=\frac{d}{dt}v(t)\)
• Position (\(s(t)\)) to Acceleration (\(a(t)\)): Acceleration is also the second derivative of the position function with respect to time.
\(a(t)=\frac{d^{2}}{dt^{2}}s(t)\)

💡Integrals (Going Backward)
• Acceleration (\(a(t)\)) to Velocity (\(v(t)\)): To find the velocity from acceleration, you take the antiderivative (integral) of the acceleration function and add a constant of integration.
\(v(t)=\int a(t)\,dt\)
• Velocity (\(v(t)\)) to Position (\(s(t)\)): To find the position from velocity, you take the antiderivative (integral) of the velocity function and add a constant of integration, which represents the initial position.
\(s(t)=\int v(t)\,dt\)

💡In Summary
• Position: Where an object is located.
• Velocity: How fast the object is changing its position and in what direction.
• Acceleration: How fast the object is changing its velocity.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link

💡Chapters:
00:00 Connecting position, velocity and acceleration
01:59 Worked example

🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials.

🔔Subscribe: https://rumble.com/user/drofeng
_______________________
⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g
_______________________
💥 Follow us on Social Media 💥
🎵TikTok: https://www.tiktok.com/@drofeng?lang=en
𝕏: https://x.com/DrOfEng
🥊: https://youtube.com/@drofeng