Hey it's a me again @drifter1! Today we continue with Physics and more specifically the branch "Classical Mechanics" to get into Impulse. So, without further ado, let's get straight into it!
ImpulseIn some of the previous articles of this series, we talked about Newton's second law and how the change in momentum is somewhat "hidden" inside of it. But, that is not the only quantity that we can derive from it. Until now we only saw that the change in momentum over time is equal to the applied force, and that alone is really useful in many calculations. What if we take the time and put it together with the force? Well, doing so we end up with:
The quantity F • dt, which is equal to the change in momentum Δp, is called Impulse and symbolised using the capital letter J. The S.I. unit of Impulse is the Newton • Second (N • s). Being a vector quantity it goes in the same direction as the Force acting on the object. So, to summarize what we said until now, we can write:
Ok, so how is this new quantity useful? Well, impulse is very useful and important in real world applications where the force is not constant. It helps us analyze problems where the forces that act on an object's motion change over time. In the end, we don't simply multiply force with time, but find the area (integral) under a force-time curve. The overall net impulse will be equal to the total change in momentum.
So, even more generalized the so called Impulse-Momentum Theorem is:
But, note that all this can be applied to constant and "changing" mass, something that is applied in rocket propulsion physics! So, Impulse can be equal to:
For any given collision we can "play" with the Impact Force F and Collision Time t, keeping the Impulse and so momentum change the same, but reducing the overall "impact" from the collision.
Similarly, the work required for any given change in kinetic energy stays constant, but we can reduce the impact force extending the collision distance:
After using Newton's laws in so many equations, I guess that you can understand how Newton's laws are connected with momentum and impulse. More specifically we can write a table such as:
|2nd law||Force law
(J = Δp)
(+F1 = -F2)
|Conservation of Momentum
(Σp = Σp')
Example from SciencenotesA 50 kg mass is sitting on a friction-less surface. An unknown constant force pushes the mass for 2 seconds until the mass reaches a velocity of 3 m/s. We have the following questions:
- What is the initial momentum of the mass?
- What is the final momentum of the mass?
- What was the force acting on the mass?
- What was the impulse acting on the mass?
1. Initial MomentumKnowing that the object it at rest at first, the initial velocity is 0 m/s. Momentum is mass times velocity and so the initial momentum is:
2. Final MomentumThe final velocity of the object after the unknown force acts on it is 3m/s,., therefore the final momentum is:
3. Acting ForceUsing Newton's second law, we can easily find the applied Force, as the change in momentum over time is equal to that Force. We know all the needed quantities, which are the change in momentum and the time that this change took place and so:
4. Acting ImpulseThe Impulse is equal to the constant Force F times the time that this force acted on the object. Therefore, the impulse is calculated to be:
This example was quite simple. Things can get more difficult when the forces that act are not constant and when more than one objects are affected by them. Either way, similarly to momentum-only problems, we have to look at the total system, as all these theorems apply to the whole system and rarely to single objects, in real world problems.
Mathematical equations used in this article, where made using quicklatex.
Previous articles of the series
- Velocity and acceleration in a rectlinear motion -> velocity, accelaration and averages of those
- Rectlinear motion with constant accelaration and free falling -> const accelaration motion and free fall
- Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
- Rectlinear motion exercises -> examples and tasks in rectlinear motion
- Position, velocity and acceleration vectors in a plane motion -> position, velocity and accelaration in plane motion
- Projectile motion as a plane motion -> missile/bullet motion as a plane motion
- Smooth Circular motion -> smooth circular motion theory
- Plane motion exercises -> examples and tasks in plane motions
Newton's laws and Applications
- Force and Newton's first law -> force, 1st law
- Mass and Newton's second law -> mass, 2nd law
- Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
- Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
- Contact forces and friction -> contact force, friction
- Dynamics of Circular motion -> circular motion dynamics, applications
- Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
- Contact force and friction examples -> exercises in force and friction
- Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
- Advanced Newton law examples -> advanced (more difficult) exercises
Work and Energy
- Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
- Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
- Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
- Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
- Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
- Potential Energy Diagrams -> Internal Energy, Internal Work
Momentum and Impulse
- Conservation of Momentum -> Momentum, Conservation of Momentum
- Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
- Collision Examples -> Various Elastic and Inelastic Collision Examples
Final words | Next up
This is actually it for today's post! Next time we will get into the motion of the Center of Mass..
Keep on drifting!