Integrating non-constant acceleration to give time

In summary, the conversation is about a personal project related to Physics, involving an object accelerating due to a force and experiencing friction. The result for velocity is expected to reach a certain value as time increases. The next step involves integrating acceleration with respect to time, but the person is unsure due to lack of confidence in integration and a cold. The equations involved include the accelerating force, friction force, total force, acceleration, and velocity. The person's attempts at integration result in a 3rd degree polynomial which does not lead to a limit for large values. They are unsure of what they are missing and have tried various substitutions and rearranging the equation.
  • #1
Danny252
1
0

Homework Statement


Not specifically a homework assignment, but for a personal project - but it's almost entirely parallel with my Physics course at the moment, and is mostly a homework-style question!

I have an object acclerating due to a force, experiencing friction. Both the accelerating and friction forces depend on velocity.

I would expect the result for v to tend to a certain value as time increases, similar to terminal velocity.

I know this should involve integrating acceleration with respect to time - but the combination of questionable integration confidence and a cold mean I just can't fathom the next step.

Homework Equations


Accelerating force = [tex]k/v[/tex] (decreases as v increases)
Friction force = [tex]a + bv + cv^{2}[/tex] (increases as v increases)
Total force = [tex]k/v - (a + bv + cv^{2})[/tex]
Acceleration = [tex]\sum F/m[/tex]
Velocity = [tex]\int a = \int (k/v - (a + bv + cv^2))/m[/tex]

The Attempt at a Solution


My attempts at integration end with a 3rd degree polynomial:
[tex](k\;ln(v) - (av + bv^2/2 + cv^3/3))/m[/tex]
Whereas I expect t in the equation, and this does not lead to a limit (for sufficiently large values it gives negative speed).
 
Physics news on Phys.org
  • #2
I'm sure that I'm missing something, but I just can't see what it is. I've tried various substitutions and re-arranging the equation, but to no avail.
 

1. How do you calculate time given non-constant acceleration?

To calculate time given non-constant acceleration, you can use the formula t = ∫(1/a)dv, where t is time, a is acceleration, and v is velocity. This formula is derived from the definition of acceleration as the rate of change of velocity over time.

2. Can you give an example of integrating non-constant acceleration to give time?

Sure, let's say an object has an initial velocity of 10 m/s and an acceleration of 2 m/s². To find the time it takes for the object to reach a velocity of 20 m/s, we can use the formula t = ∫(1/2)dv. Integrating this equation gives us t = (1/2)v + C, where C is a constant. Plugging in the given values, we get t = (1/2)(20) + C = 10 + C. Since we know the initial velocity is 10 m/s, we can solve for C and get a final time of 10 seconds.

3. What is the difference between constant and non-constant acceleration?

Constant acceleration refers to a situation where the acceleration remains the same throughout the motion of an object. This means that the velocity of the object changes at a constant rate. Non-constant acceleration, on the other hand, means that the acceleration is changing over time, resulting in a changing velocity of the object.

4. Can you integrate acceleration without knowing the initial velocity?

No, in order to integrate acceleration to find time, you need to know the initial velocity of the object. This is because the integration process involves using the initial velocity to determine the constant of integration, which is necessary for finding the final time value.

5. What are some real-life applications of integrating non-constant acceleration to give time?

Integrating non-constant acceleration to give time has many practical applications, such as calculating the time it takes for a rocket to reach a certain velocity, determining the time it takes for a car to reach a certain speed, or predicting the time it takes for a rollercoaster to complete a loop. It is also used in engineering and physics to analyze and design various types of motion, such as projectile motion and circular motion.

Similar threads

Replies
30
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
974
  • Introductory Physics Homework Help
Replies
3
Views
696
  • Introductory Physics Homework Help
Replies
2
Views
783
  • Introductory Physics Homework Help
Replies
29
Views
823
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
2
Replies
69
Views
4K
  • Introductory Physics Homework Help
Replies
11
Views
3K
  • Introductory Physics Homework Help
Replies
1
Views
761
  • Introductory Physics Homework Help
3
Replies
97
Views
3K
Back
Top