Acceleration in Kinematics: How Does it Vary with Displacement?

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SUMMARY

The discussion centers on determining how acceleration varies with displacement for a particle moving along a straight line, described by the displacement function x = sqrt(at + 2bt + c), where a, b, and c are constants. Participants clarify that the task is to express acceleration in terms of displacement, emphasizing the need to derive the second derivative of x with respect to time to find acceleration. The method involves using the chain rule, specifically dz/dx = (dz/dt)/(dx/dt), to relate acceleration to displacement. This approach, while complex, is confirmed as technically correct by multiple contributors.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and derivatives.
  • Familiarity with kinematic equations and their applications.
  • Knowledge of chain rule in calculus.
  • Basic grasp of particle motion in one-dimensional space.
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  • Study the process of taking derivatives of functions, focusing on second derivatives.
  • Learn about the chain rule in calculus and its applications in physics.
  • Explore kinematic equations and their derivations in detail.
  • Practice problems involving acceleration and displacement relationships in particle motion.
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Students studying physics, particularly those focusing on kinematics, as well as educators and tutors looking to clarify concepts related to acceleration and displacement in motion.

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Homework Statement



A particle moves along a straight line such that its displacement x changes with time t as x= sqrt( at+2bt + c) where a, b and c are constants, then the acceleration varies with x as

Homework Equations





The Attempt at a Solution


I can't figure out how do i solve it. Using graphs as well, I could not arrive at any conclusion.
 
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The question is incomplete. Please post the full question
 
No, @adjacent, the question isn't incomplete. We have to fill in the blank. "the acceleration varies with x as ________ ". Sorry for not mentioning it clearly!
 
ritik.dutta3 said:
No, @adjacent, the question isn't incomplete. We have to fill in the blank. "the acceleration varies with x as ________ ". Sorry for not mentioning it clearly!

When you are given the position as a function of time, how do you find the acceleration using derivative?
 
My suggestion:
First take second derivative of x with respect to t to get acceleration which we will call z.
Then use dz/dx = (dz/dt)/(dx/dt)

Looks messy, but technically correct. In the final result, substitute for x where it appears.
 

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