Momentum
Momentum → inertia in motion.
Momentum of an object is the product of the mass and velocity of the object.
Momentum = mass x velocity (mv)
Without considering direction, momentum = mass x speed
It’s harder to stop a moving truck than a moving car, because the truck has more momentum than the car.
The larger the mass or velocity, the larger the momentum.
Momentum changes with changes in either mass or velocity or both.
A change in velocity causes acceleration.
Acceleration is caused by force.
Changes in momentum also depends on time → how long the force is applied.
If you apply force to a stalled automobile, there is a change in its momentum.
If you apply the force for a larger amount of time, there is a greater change in the momentum.
Impulse = force x time interval
Impulse = Ft
Impulse = change in momentum
Shorthand notation: Ft = Δ(mv)
Δ → delta, m → mass, v → velocity
Impulse-Momentum relationship → Impulse can cause a change in momentum, change in momentum can cause impulse.
Momentum is increased by applying the greatest possible force for the longest possible time.
The forces involved vary from instant to instant, so the average force is considered overall.
Consider a golf club striking a ball.
Zero force is exerted on the ball until contact occurs.
Once contact happens, the force increases rapidly.
The ball gets distorted at this point.
As the ball comes back into shape, the force decreases.
Impulse is a product of force and time.
The momentum can be brought to zero over either a shorter or a longer period of time, with a lesser force.
Whenever we want the force to be smaller, we increase the time during which the momentum decreases to zero.
Eg. 1: A car out of control. You can stop it by driving it either into a haystack or into a wall. You choose the haystack.
Eg. 2: Catching a baseball. You stretch your arm out to catch it to allow yourself to move backwards.
Eg. 3: Jumping from an elevated surface. When you do so, you bend your knees when you land.
In each of these cases, the time over which the momentum becomes zero is increased, and the force is thus decreased.
For an object being brought to rest, if the time is shorter, the force is larger.
Eg. 1: A karate expert splitting a stack of bricks in a short amount of time.
Eg. 2: A boxer moving into a punch instead of away from it.
In each of these cases, the time over which the momentum becomes zero is short, so the force is much larger.
In the case of the karate expert, if the hand bounces on impact, the force is even greater.
It takes a larger impulse to throw an object back the way it came than to just stop it.
When you stop an object, you bring its momentum to zero.
When you throw an object back, you have to provide extra impulse.
Eg: Consider swinging a dart at a wooden block.
When the dart has a nail at the edge of it, it stops as it sticks to the block, and the block stays upright.
When the nail is removed, the dart bounces back instead of sticking to the block, and the block topples over.
Reason → When the dart bounces against the wooden block, the force is greater.
Only an external impulse can change the momentum of a system.
Internal forces and impulses don’t work.
Eg: Molecular forces within a baseball don’t affect the momentum of the ball in any sort of way.
Reason → Internal forces always come in balanced pairs.
Consider a cannon being fired from a cannonball.
Force on the cannonball in the cannon = Force that causes cannon to recoil.
The forces act for the same amount of time.
The impulses are equal and opposite.
Newton’s third law applies to impulses too.
After firing the cannonball, the net momentum is still zero. Momentum is neither gained nor lost.
Momentum is a vector quantity.
Vector quantity → has both magnitude and direction.
The momenta in a system can cancel out, and the overall momentum can be equal to zero.
Without any net force or net impulse on a system, the momentum of that system can’t change.
Law of Conservation of Momentum:
In the absence of an external force, the momentum of a system remains unchanged.
Momentum is conserved in collisions.
Momentum of a system remains the same before and after a collision.
Reason → The forces in a collision are internal forces.
Net momentum before collision = Net momentum after collision
Types of collisions:
Elastic collision → No lasting deformation on the rebounding object, no generation of heat.
Inelastic collision → Lasting deformation on the rebounding object, or the generation of heat, or both.
In a perfectly inelastic collision, the objects stick together.
Consider two freight cars. One is at rest, one is moving towards the first.
If the mass of each car is m and the second car is moving at a speed of 10 m/s, then:
momentum before = momentum after
→ (m x 10) = (2m x V), where V is the speed of both after the collision
after the collision, the mass is 2m because we now consider the masses of both freight cars.
If two objects A and B are moving with equal momenta in:
opposite directions → the net momentum adds up to zero.
the same direction → the net momentum is just the addition of the individual momenta.
Consider the object A moving eastward with 10 units more momentum than B.
After they collide, both objects will move east with 10 units of momentum.
Objects that move due to collisions eventually come to rest due to friction.
During collisions, time of impact is short and force of impact is large.
Impact force is much larger than frictional force, so momentum is conserved immediately before and after the collision.
Consider docking a spacecraft.
Friction is entirely absent.
Net momentum before docking = net momentum after docking.
Momentum is preserved before and after docking.
Numerical Example:
Consider a larger fish swimming towards a smaller fish at rest and swallowing it.
Mass of the larger fish = 5kg
Mass of the smaller fish = 1kg
Velocity of the larger fish before swallowing = 1m/s
Velocity of the larger fish after swallowing = ?
(The effects of water resistance are being ignored.)
Solution:
→ Net momentum before swallowing = Net momentum after swallowing
→ 5 x 1 = (5+1) x (velocity of the larger fish after swallowing, v)
→ v = (5 x 1)/6
→ v = 5/6 m/s
Points to Note:
The smaller fish has no momentum before swallowing because it is at rest, i.e, its velocity is zero.
After swallowing the smaller fish, the combined mass of the larger and smaller fish becomes the new mass.
After swallowing, the smaller fish acquires the same velocity as the larger fish.
Variation:
Consider the same two fish. This time, however, the smaller fish is moving towards the left at a velocity of 4m/s.
→ net momentum before = net momentum after
→ (5 x 1) + (1 x (-4)) = (5 +1) x (velocity of the larger fish after swallowing, v)
→ v = (5 - 4)/6
→ v = 1/6 m/s
The minus sign in -4 shows that the smaller fish is moving at a velocity of 4m/s in a direction opposite to the direction of the larger fish.
In all collisions, the net momentum remains the same before and after the collision.
If the colliding objects are acting along different angles, the parallelogram law is used to find the net momentum.
Example 1:
Consider two cars, A and B, colliding.
Car A is moving east, car B is moving north.
Both cars have momenta equal in magnitude. 
We use the parallelogram law to find the net momentum of the two cars.
The diagonal of the square formed by the two cars gives us the net momentum.
For any square, the diagonal is √2 times the length of the side.
In this case, net momentum = √2 times the momentum of either car.
Example 2:
Consider a falling firecracker exploding into two pieces.
You can combine the momenta of the fragments and find the original momentum of the firecracker.
Whatever happens during a collision, the net momentum before the collision will always be equal to the net momentum after the collision.
Momentum → inertia in motion.
Momentum of an object is the product of the mass and velocity of the object.
Momentum = mass x velocity (mv)
Without considering direction, momentum = mass x speed
It’s harder to stop a moving truck than a moving car, because the truck has more momentum than the car.
The larger the mass or velocity, the larger the momentum.
Momentum changes with changes in either mass or velocity or both.
A change in velocity causes acceleration.
Acceleration is caused by force.
Changes in momentum also depends on time → how long the force is applied.
If you apply force to a stalled automobile, there is a change in its momentum.
If you apply the force for a larger amount of time, there is a greater change in the momentum.
Impulse = force x time interval
Impulse = Ft
Impulse = change in momentum
Shorthand notation: Ft = Δ(mv)
Δ → delta, m → mass, v → velocity
Impulse-Momentum relationship → Impulse can cause a change in momentum, change in momentum can cause impulse.
Momentum is increased by applying the greatest possible force for the longest possible time.
The forces involved vary from instant to instant, so the average force is considered overall.
Consider a golf club striking a ball.
Zero force is exerted on the ball until contact occurs.
Once contact happens, the force increases rapidly.
The ball gets distorted at this point.
As the ball comes back into shape, the force decreases.
Impulse is a product of force and time.
The momentum can be brought to zero over either a shorter or a longer period of time, with a lesser force.
Whenever we want the force to be smaller, we increase the time during which the momentum decreases to zero.
Eg. 1: A car out of control. You can stop it by driving it either into a haystack or into a wall. You choose the haystack.
Eg. 2: Catching a baseball. You stretch your arm out to catch it to allow yourself to move backwards.
Eg. 3: Jumping from an elevated surface. When you do so, you bend your knees when you land.
In each of these cases, the time over which the momentum becomes zero is increased, and the force is thus decreased.
For an object being brought to rest, if the time is shorter, the force is larger.
Eg. 1: A karate expert splitting a stack of bricks in a short amount of time.
Eg. 2: A boxer moving into a punch instead of away from it.
In each of these cases, the time over which the momentum becomes zero is short, so the force is much larger.
In the case of the karate expert, if the hand bounces on impact, the force is even greater.
It takes a larger impulse to throw an object back the way it came than to just stop it.
When you stop an object, you bring its momentum to zero.
When you throw an object back, you have to provide extra impulse.
Eg: Consider swinging a dart at a wooden block.
When the dart has a nail at the edge of it, it stops as it sticks to the block, and the block stays upright.
When the nail is removed, the dart bounces back instead of sticking to the block, and the block topples over.
Reason → When the dart bounces against the wooden block, the force is greater.
Only an external impulse can change the momentum of a system.
Internal forces and impulses don’t work.
Eg: Molecular forces within a baseball don’t affect the momentum of the ball in any sort of way.
Reason → Internal forces always come in balanced pairs.
Consider a cannon being fired from a cannonball.
Force on the cannonball in the cannon = Force that causes cannon to recoil.
The forces act for the same amount of time.
The impulses are equal and opposite.
Newton’s third law applies to impulses too.
After firing the cannonball, the net momentum is still zero. Momentum is neither gained nor lost.
Momentum is a vector quantity.
Vector quantity → has both magnitude and direction.
The momenta in a system can cancel out, and the overall momentum can be equal to zero.
Without any net force or net impulse on a system, the momentum of that system can’t change.
Law of Conservation of Momentum:
In the absence of an external force, the momentum of a system remains unchanged.
Momentum is conserved in collisions.
Momentum of a system remains the same before and after a collision.
Reason → The forces in a collision are internal forces.
Net momentum before collision = Net momentum after collision
Types of collisions:
Elastic collision → No lasting deformation on the rebounding object, no generation of heat.
Inelastic collision → Lasting deformation on the rebounding object, or the generation of heat, or both.
In a perfectly inelastic collision, the objects stick together.
Consider two freight cars. One is at rest, one is moving towards the first.
If the mass of each car is m and the second car is moving at a speed of 10 m/s, then:
momentum before = momentum after
→ (m x 10) = (2m x V), where V is the speed of both after the collision
after the collision, the mass is 2m because we now consider the masses of both freight cars.
If two objects A and B are moving with equal momenta in:
opposite directions → the net momentum adds up to zero.
the same direction → the net momentum is just the addition of the individual momenta.
Consider the object A moving eastward with 10 units more momentum than B.
After they collide, both objects will move east with 10 units of momentum.
Objects that move due to collisions eventually come to rest due to friction.
During collisions, time of impact is short and force of impact is large.
Impact force is much larger than frictional force, so momentum is conserved immediately before and after the collision.
Consider docking a spacecraft.
Friction is entirely absent.
Net momentum before docking = net momentum after docking.
Momentum is preserved before and after docking.
Numerical Example:
Consider a larger fish swimming towards a smaller fish at rest and swallowing it.
Mass of the larger fish = 5kg
Mass of the smaller fish = 1kg
Velocity of the larger fish before swallowing = 1m/s
Velocity of the larger fish after swallowing = ?
(The effects of water resistance are being ignored.)
Solution:
→ Net momentum before swallowing = Net momentum after swallowing
→ 5 x 1 = (5+1) x (velocity of the larger fish after swallowing, v)
→ v = (5 x 1)/6
→ v = 5/6 m/s
Points to Note:
The smaller fish has no momentum before swallowing because it is at rest, i.e, its velocity is zero.
After swallowing the smaller fish, the combined mass of the larger and smaller fish becomes the new mass.
After swallowing, the smaller fish acquires the same velocity as the larger fish.
Variation:
Consider the same two fish. This time, however, the smaller fish is moving towards the left at a velocity of 4m/s.
→ net momentum before = net momentum after
→ (5 x 1) + (1 x (-4)) = (5 +1) x (velocity of the larger fish after swallowing, v)
→ v = (5 - 4)/6
→ v = 1/6 m/s
The minus sign in -4 shows that the smaller fish is moving at a velocity of 4m/s in a direction opposite to the direction of the larger fish.
In all collisions, the net momentum remains the same before and after the collision.
If the colliding objects are acting along different angles, the parallelogram law is used to find the net momentum.
Example 1:
Consider two cars, A and B, colliding.
Car A is moving east, car B is moving north.
Both cars have momenta equal in magnitude.
We use the parallelogram law to find the net momentum of the two cars.
The diagonal of the square formed by the two cars gives us the net momentum.
For any square, the diagonal is √2 times the length of the side.
In this case, net momentum = √2 times the momentum of either car.
Example 2:
Consider a falling firecracker exploding into two pieces.
You can combine the momenta of the fragments and find the original momentum of the firecracker.
Whatever happens during a collision, the net momentum before the collision will always be equal to the net momentum after the collision.
