The position-time relationship for a moving object is given...

Name: Problem 15, Chapter 3: Nelson Calculus and Vectors
Uploaded: 2021-06-15T21:08:03-08:00
Duration: 2 min 43 s
Channel: Lucas Finney
Description: The position-time relationship for a moving object is given by s(t)=k t^2+(6 k^2-10 k) t+2 k, where k is a non-zero constant. a. Show that the acceleration is constant. b. Find the time at which the velocity is zero, and determine the position of the object when this occurs.

The position-time relationship for a moving object is given by $s(t)=k t^{2}+\left(6 k^{2}-10 k\right) t+2 k,$ where $k$ is a non-zero constant. a. Show that the acceleration is constant. b. Find the time at which the velocity is zero, and determine the position of the object when this occurs.

The position-time relationship for a moving object is given by $s(t)=k t^{2}+\left(6 k^{2}-10 k\right) t+2 k,$ where $k$ is a non-zero constant.
a. Show that the acceleration is constant.
b. Find the time at which the velocity is zero, and determine the position of the object when this occurs.

Key Concepts

Quadratic Functions

Quadratic functions are polynomials of degree two and are common in the study of motion, especially when acceleration is constant. Their well-known properties, such as symmetrical graphs and calculable roots, make them instrumental in finding key points in the motion analysis, like the time when velocity becomes zero.

Critical Points in Motion

In motion analysis, setting the velocity function (the first derivative of the position function) equal to zero allows one to find critical points, such as when the moving object momentarily stops. By solving for the time at which the velocity is zero and substituting back into the position function, one can determine the corresponding spatial location.

Derivatives in Kinematics

In kinematics, the instantaneous rates of change are crucial. The first derivative of the position function with respect to time gives the velocity, while the second derivative gives the acceleration. This process, based on differentiation, allows one to move from a displacement description to dynamic quantities that describe motion.

Constant Acceleration

An object is said to have constant acceleration if the acceleration, given by the second derivative of the position function, does not depend on time. Showing that the acceleration is constant involves differentiating the quadratic position function twice and verifying that the resulting expression is independent of time, which is central to the analysis of uniformly accelerated motion.

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