Force and momentum
Newton’s Second Law gives us a way to calculate how much an object will accelerate when a force is applied to it. It says:
But also acceleration is the change in velocity over time as we have already studied before.
So combining both these formulae we get,
As mv is final momentum and mu is initial momentum.
here J shows impulse.
So you can say that force is equal to change in momentum over time.
Impulse
Definition:
Impulse is the change in momentum of an object when a force is applied over a period of time.
Formula:
Mathematically, it is expressed as the product of force and time
F × t = Δ (mv)
F × t = Δ p
Or
F × t = J
Where,
F = force
t= time
j= impulse
Δp = change in momentum
Δ (mv) = change in momentum
Units:
Measured in Newton-seconds (Ns).
Explanation:
Impulse is a measure of how much an object’s momentum changes when a force acts on it for a certain amount of time. Momentum is the product of an object’s mass and its velocity (momentum=mass×velocity). When a force is applied to an object, it changes the object’s velocity, thereby changing its momentum. The amount of this change is the impulse.
Impulse as a vector quantity
Impulse is a vector quantity, meaning it has both magnitude and direction. It shares the same direction as the force applied.
The impulse experienced by an object is directly proportional to both the magnitude of the force applied (also called impact force) and the duration over which the force acts as evident from its formula. A larger force applied over a longer time interval will result in a greater impulse, causing a more significant change in momentum.
Graphically, the impulse is represented by the area under a force vs. time graph. For a constant force, this area corresponds to the product of force and time.
Analogy:
Imagine you’re throwing a ball.
Impulse: Think of impulse as the “push” you give the ball when you throw it.
Force: Your hand applies a force to the ball.
Time: This force is applied over a short period of time while your hand is in contact with the ball.
The combination of this force and the time it acts is the impulse.
Resultant Force/Impact Force as Change in Momentum per Unit Time:
Definition: Impact force is the force exerted during a collision or impact between two objects.
Relation to Impulse:
The impact force multiplied by the duration of the impact equals the impulse.
F × t = J
Or
F= J/t = Δp/t
Thus, impulse is the cause of the change in momentum, while the impact force is the immediate cause of this change.
Imagine you need to move a stationary car. Here’s how impulse and impact force can be understood through this scenario:
Impact Force: When you start pushing the car, you apply a force. This force is strong but acts for a short period of time.
Think of the impact force as the initial push you give the car to start it moving. This is when you put in a lot of effort to overcome the car’s inertia (resistance to motion). This initial push represents the impact force.
Impulse: Impulse is the overall effect of the force applied over the time it is exerted. It’s the product of the impact force and the duration for which the force is applied.
Impulse=Force×Time
Imagine you keep pushing the car steadily for a few seconds. The continuous push (force applied over time) changes the car’s motion, making it start to roll. The impulse is the combination of the force you apply and the time you keep applying it.
Formula:
Impact Force= Impulse Time or
𝐹 = Δ 𝑝 / Δ 𝑡
Units: Measured in Newtons (N).
Explanation
When an object collides with another, a large force is exerted over a short period of time. This force is the impact force. The change in momentum (impulse) during the collision is a result of this impact force.
Analogies
Bouncing Ball:
When you throw a ball at a wall, it bounces back. The wall exerts an impact force on the ball, changing its direction and speed. The impulse is the change in the ball’s momentum due to the force exerted by the wall over the brief contact time.
Airbag in a Car:
When a car suddenly stops due to a collision, the passengers continue moving forward due to inertia. An airbag inflates and increases the time over which the passengers’ momentum changes (decelerates). By increasing this time, the impact force is reduced, lessening injuries.
Solved Examples
Example 1: Calculating Impulse
Problem: A 5 kg ball is moving at 10 m/s. It is hit by a bat, causing it to reverse direction and move at 15 m/s. Calculate the impulse delivered by the bat.
Solution:
Example 2: Calculating Impact Force
Problem: A 0.1 kg tennis ball moving at 20 m/s hits a wall and comes to rest in 0.05 seconds. Calculate the impact force exerted by the wall.
Solution:
Resultant force describes the impact on an object’s momentum. It determines whether the object speeds up, slows down, or changes direction based on how the force affects its motion.
Conservation of Momentum:
In a closed system where no external forces act, the total momentum before and after a collision or interaction remains constant.
This principle is known as the conservation of momentum and is a powerful tool in analysing collisions and interactions between objects.
Law of Conservation of Momentum
The total momentum of an isolated system remains constant if no external forces act on it
In essence, momentum encapsulates an object’s mass and velocity into a single quantity that describes its motion. The greater the mass or the faster the velocity, the greater the momentum. Momentum plays a pivotal role in describing and predicting the motion of objects. Let’s delve into the key aspects of the conservation of momentum:
The conservation of momentum applies to isolated systems. An isolated system is one in which the net external force acting on the system is zero. This means that the system is not influenced by external factors like friction, air resistance, or other forces from outside the system.
The conservation of momentum principle states that if the net external force acting on an isolated system is zero, then the total momentum of the system before an event (like a collision or explosion) is equal to the total momentum after the event.
Total Initial Momentum = Total Final Momentum
It can be mathematically expressed using equations. For a two-object system (in one dimension):
m1 . u1 + m2 . u2 = m1 . v1 + m2 . v2
Here, m1 and m2 are the masses of the two objects
u1 and u2 are their initial velocities,
and v1 and v2 are their final velocities.
Analogy:
Imagine two toy cars racing towards each other on a track. One car is light and speeding, while the other is heavy and slower. When they collide:
Momentum:
Momentum is like the push each car has. The faster, lighter car has more momentum because it’s moving faster.
Collision Effect:
When they crash, their total momentum stays the same, but it’s divided differently between them.
Outcome:
The fast car might bounce back quickly, transferring some of its momentum to the slower car, which keeps moving forward steadily.
In this collision, momentum helps explain how objects interact when they collide, showing how energy and motion are transferred between them.
