Classical mechanics (find trajectory and kinetic energy)

In summary, the problem asks to find the kinetic energy and trajectory of a particle at the point x=2, given a force function of -12x+6, with initial conditions of position and velocity both equal to 0. The two suggested approaches are to find the potential function using the fact that the curl of the force is 0, or to solve the differential equation using a change of variable to eliminate the constant. This problem belongs in the Introductory Physics homework section.
  • #1
NanoMath
11
0

Homework Statement


Given the force ## \vec{ F }(x) = (-12x + 6) \vec{i} ## ; find kinetic energy ## T## at the point ##x=2## and trajectory of a particle ## \vec{r}(t) ##, given that ## \vec{r}(t=0)=\vec{0}## and ##\dot{\vec{r}}(t=0)=\vec{0}## .

3. The Attempt at a Solution

Since ##\nabla\times \vec{F} = \vec{0}## we can find potential function ## \vec{F}(x) = -\nabla U(x)## , therefore ## U(x)= 6x²-6x + C ## . From here I am not sure how to proceed.
Second idea is to solve differential equation but I'm not sure how to do it because force is a function of ## x## and not ## t ## .
$$ m\ddot{x}=-12x+6$$ $$\ddot{x}+\frac{1}{m}12x-\frac{1}{m}6=0$$ let's suppose ##m=1## for simplicity, then we get $$ \ddot{x}+12x-6=0$$
 
Physics news on Phys.org
  • #2
NanoMath said:

Homework Statement


Given the force ## \vec{ F }(x) = (-12x + 6) \vec{i} ## ; find kinetic energy ## T## at the point ##x=2## and trajectory of a particle ## \vec{r}(t) ##, given that ## \vec{r}(t=0)=\vec{0}## and ##\dot{\vec{r}}(t=0)=\vec{0}## .

3. The Attempt at a Solution

Since ##\nabla\times \vec{F} = \vec{0}## we can find potential function ## \vec{F}(x) = -\nabla U(x)## , therefore ## U(x)= 6x²-6x + C ## . From here I am not sure how to proceed.
Second idea is to solve differential equation but I'm not sure how to do it because force is a function of ## x## and not ## t ## .
$$ m\ddot{x}=-12x+6$$ $$\ddot{x}+\frac{1}{m}12x-\frac{1}{m}6=0$$ let's suppose ##m=1## for simplicity, then we get $$ \ddot{x}+12x-6=0$$
With a simple change of variable you can eliminate the constant 6 from the differential equation and obtain a very well known form. Think springs.
By the way, this belongs in Introductory Physics homework.
 

1. How do you find the trajectory of an object in classical mechanics?

In classical mechanics, the trajectory of an object can be determined by using the equations of motion, specifically the position, velocity, and acceleration equations. These equations take into account the initial conditions and the forces acting on the object, such as gravity or friction. By solving for the position equation, the trajectory of the object can be plotted on a graph.

2. What factors affect the trajectory of an object in classical mechanics?

The trajectory of an object in classical mechanics is affected by various factors, such as the initial velocity and position of the object, the forces acting on it, and the mass of the object. Other factors that can affect the trajectory include air resistance, gravity, and any external forces, such as friction from a surface.

3. How is kinetic energy calculated in classical mechanics?

Kinetic energy in classical mechanics is calculated using the equation KE = 1/2 * mv^2, where m is the mass of the object and v is the velocity. This equation represents the energy an object possesses due to its motion. Kinetic energy is a scalar quantity and is measured in joules (J).

4. What is the relationship between kinetic energy and velocity in classical mechanics?

In classical mechanics, kinetic energy is directly proportional to the square of an object's velocity. This means that as the velocity of an object increases, its kinetic energy also increases. Additionally, if the mass of the object remains constant, a doubling of the velocity will result in a quadrupling of the kinetic energy.

5. Can classical mechanics be used to accurately predict the trajectory and kinetic energy of all objects?

Classical mechanics is a mathematical model that provides accurate predictions for the trajectory and kinetic energy of objects in most everyday situations. However, in extreme conditions, such as at very high speeds or in the presence of strong gravitational fields, the principles of classical mechanics may not apply and more advanced theories, such as relativity and quantum mechanics, may be needed.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
182
  • Introductory Physics Homework Help
Replies
3
Views
183
  • Introductory Physics Homework Help
Replies
4
Views
511
  • Introductory Physics Homework Help
Replies
15
Views
300
  • Introductory Physics Homework Help
Replies
25
Views
207
  • Introductory Physics Homework Help
Replies
3
Views
747
  • Introductory Physics Homework Help
Replies
3
Views
136
  • Introductory Physics Homework Help
Replies
1
Views
262
  • Introductory Physics Homework Help
Replies
14
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
178
Back
Top