A 4.00 -kg particle moves along the x axis. Its position...

A 4.00 -kg particle moves along the $x$ axis. Its position varies with time according to $x=t+2.0 t^{3},$ where $x$ is in meters and $t$ is in seconds. Find $(a)$ the kinetic energy of the particle at any time $t,$ (b) the acceleration of the particle and the force acting on it at time $t,(c)$ the power being delivered to the particle at time $t,$ and $(\mathrm{d})$ the work done on the particle in the interval $t=0$ to $t=2.00 \mathrm{s}$ .

Key Concepts

Power

Power in physics is defined as the rate at which work is done or energy is transferred over time. It can also be determined as the product of force and velocity when the force is applied in the direction of motion. Understanding power is essential when analyzing how energy is continuously being input or converted in a system.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and acceleration. This principle is fundamental for understanding how forces result in changes in an object's motion. It provides the connection between the derived acceleration from kinematics and the dynamic force experienced by an object.

Work and Energy Theorem

The work and energy theorem states that the work done on an object is equal to the change in its kinetic energy. Work, calculated as the integral of force over displacement, provides a bridge between dynamics and energy. This concept is crucial for understanding how forces acting over distances change the energy state of a system.

Kinematics and Differentiation

Kinematics involves describing the motion of objects using parameters such as displacement, velocity, and acceleration. In problems where the position is given as a function of time, finding the velocity and acceleration requires differentiation with respect to time. This process allows us to relate the motion to other physical quantities through their time derivatives.

Kinetic Energy

Kinetic energy is a form of energy that a body possesses due to its motion. It is defined as one half of the product of mass and the square of velocity. This concept is central in mechanics, linking the state of motion of an object with energy considerations in various physical processes.

Transcript

00:01 In this problem, we're going to talk about several topics concerning one -dimensional motion.

00:06 So let's say that we have a particle that is undergoing one -dimensional motion, where the x -coronid is given by a certain function x of t.

00:18 Then we can calculate the velocity of that particle as d x d t, the acceleration as dv -tt, or d t.

00:31 2x d t2 that is the second derivative of x also the force applied on that object is equal to the mass of the object times the acceleration the kinetic energy of that object is the mass times v squared over 2 and also we need to know that the power is the derivative of the energy with respect to time.

01:11 Okay, so these are the concepts we're going to use in our problem.

01:15 So basically, what we have is an object that has a mass of four kilograms that is traveling with a trajectory that is given by a function x of t, that is equal to t plus 2t, q.

01:35 And in question a, our goal is to find the kinetic energy of the time, particle as a function of t so what i'm going to do first is to calculate v and v is d x d t so it's the derivative of t plus two t cubed the derivative of t is one the derivative of t cubed is three t squared so three times two six so uh the kinetic energy is the mass times v squared over two the mass is four then we divide it by 4 and multiply it by 1 plus 6 t squared squared.

02:21 So the kinetic energy is 2 times 1 plus 6 t squared squared in joules.

02:36 In question b, we have to calculate both the force and the acceleration on that object.

02:46 Let's start with the acceleration.

02:48 Now we know that the acceleration is a derivative of the velocity with respect to 10.

02:52 The velocity is 1 plus 6 t squared.

03:00 The derivative of 1 is 0.

03:03 The derivative of t squared is 2t.

03:06 So the acceleration is 12 t.

03:09 And the force is just the mass times the acceleration.

03:14 So it's 4 kilograms times 12t, so that's 48 t joules...