Linear Momentum Impulse Collisions

1. Linear Momentum

1.1 Linear Momentum Definition

The linear momentum of an object is defined as the result of multiplying the mass of the object by the velocity of the object.

p = m v

where:

p = momentum, m = mass (kg), v = velocity (m/s)

Linear momentum, or simply momentum, is a vector quantity as it is derived by multiplying a vector (velocity) and a scalar (mass). As momentum is a vector quantity, it has direction and magnitude. Momentum shares direction with the velocity or motion of an object.

Momentum is proportional to mass and velocity since the greater the mass, the greater the momentum. Likewise, the greater the velocity, the greater the momentum. Suppose there are two cars, say cars A and B. If car A’s mass is greater than car B’s and both cars move at the same velocity, car A will have greater momentum than that of car B. Similarly, if cars A and B are of the same mass, but car A moves faster than car B, car A’s momentum is greater than that of car B.

If an object that has mass does not move or is at rest (has zero velocity), the momentum of the object is zero.

The SI unit of momentum is kg m/s, which is comprised of the unit of mass and unit of velocity.

1.2 Newton’s Second Law

Previously, you have learned Newton’s Second Law which is stated in the equation ΣF = m a and explains the relationship between the net force and mass as well as acceleration of an object. The net force acting on an object which has mass renders acceleration to the object. This time, you are to be introduced to another form of Newton’s Second Law, which explains the relationship between the net force and change in momentum of an object.

If the net force acts on an object which is initially at rest, the object will move. Before moving, the object does not have any momentum. The object has momentum after movement is rendered. In other words, the net force acting on the object causes a change in the object’s momentum for a given time interval. The rate of change in an object’s momentum is equal to the net force acting on the object.

Where:

ΣF = net force (Newton), Δt = time interval (second), Δp = m (v_{t} – v_{o}) = change in momentum (kg m/s).

Equation 1.1 is another form of Newton’s Second Law, which explains the relationship between the net force and rate of change in momentum of an object, either when the object’s mass is constant or changes.

Where:

ΣF = net force (Newton), m = mass (kg), a = acceleration (m/s^{2})

Equation 1.2 is a Newton’s Second Law equation that explains the relationship between the net force and acceleration of an object with a constant mass.

2. Impulse

2.1 Impulse Definition

Impulse is defined as the result of multiplying force or net force by the time interval.

Where:

I = impulse, ΣF = net force (Newton), Δt = time interval (second).

2.2 Impulse-Momentum Theorem

Impulse-momentum theorem is obtained by deriving an equation from equation 1.1

ΣF Δt = Δp

I = Δp ………………….. Equation 1.3

Equation 1.3 indicates that impulse is equal to change in momentum.

I = ΣF Δt

Δp = m v_{t} – m v_{o } = m (v_{t }– v_{o})

Example question 1:

A ball with a mass of 1 kg is thrown horizontally at a speed of 2 m/s. Then, the ball is hit in the same direction as the initial direction. The ball takes 1 ms to come into contact with the hitter, and the speed of the ball after leaving the hitter is 4 m/s. What is the force applied by the hitter on the ball?

Known :

mass (m) = 1 kg, Initial velocity (v_{o}) = 2 m/s, time interval (Δt) = 1 x 10^{-3 }second, final velocity (v_{t}) = 4 m/s

The direction of the ball’s motion does not change, thus the initial speed and the final speed have the same mark.

Wanted: force (F)

Solution :

Example question 2:

A ball with a mass of 1 kg is thrown horizontally to the right at a speed of 10 m/s. After being hit, the ball moves to the left at a speed of 20 m/s. Determine the impulse is acting on the ball.

Known :

mass (m) = 1 kg

Initial velocity (v_{o}) = 10 m/s,

Final velocity (v_{t}) = -20 m/s

The directions of the ball’s motion (directions of velocity) are opposite, thus the initial speed and the final speed have different sign.

Wanted: Impulse (I)

__Solution :__

I = m (v_{t} – v_{o}) = 1 kg (-20 m/s – 10 m/s) = 1 kg (-30 m/s) = – 30 kg m/s

The negative sign indicates that the direction of the impulse is the same as the direction of the final speed of the ball (to the left)

Example question 3

A student hits a 0.1 kg volleyball which is initially at rest. The student’s hand comes into contact with the volleyball for 0.01 second. After being hit, the volleyball moves at a speed of 2 m/s.

(a) What is the amount of force exerted by the student’s hand to the volleyball?

(b) Newton’s Third Law states that if the student exerts force to the volleyball, the volleyball will exert force too to the student. What is the size of force exerted by the volleyball to the student’s hand?

(c) If the student’s hand comes into contact with the volleyball for 0.001 seconds, what is the size of force exerted by the volleyball to the student’s hand?

__Known :__

mass (m) = 0.1 kg,

Time interval 1 (Δt_{1}) = 0.01 s = 1 x 10^{-2} s

Initial velocity (v_{o}) = 0

Final velocity (v_{t}) = 2 m/s

Time interval 2 (Δt_{2}) = 0.001 s = 1 x 10^{-3 }s

__Wanted:__ force (F)

__Solution :__

(a) The force applied by the student’s hand to the volleyball for a period of contact time of 0.01 second is

(b) The force exerted by the volleyball to the student’s hand for a period of contact time of 0.01 second is

Newton’s Third Law: F action = – F reaction

The size of force exerted by the ball to the student’s hand is 200 N

(c) The force exerted by the ball to the student’s hand for a period of contact time of 0.001 seconds is

Based on the results obtained, it can be concluded that the force exerted by the ball to the student’s hand is greater when the contact time is shorter. Greater force cause greater pain to the student’s hand. You can prove this when you play volleyball. The contact time you will take when you hit a harder volleyball is shorter than when you hit the softer one. The difference in the contact time makes your hand feel greater pain when you hit a harder ball.

**What is linear momentum, and how is it different from force?****Answer**: Linear momentum ($p$) of an object is the product of its mass ($m$) and its velocity ($v$), i.e., $p=mv$. While force relates to the change in momentum of an object with time, momentum itself is a measure of how much motion an object has and in what direction.

**How is impulse related to the change in momentum of an object?****Answer**: Impulse is the product of the average force applied to an object and the time duration over which it’s applied. It is equal to the change in momentum of the object. In mathematical terms: I$mpulse=Δp=F×Δt$.

**What does the conservation of momentum mean in a collision?****Answer**: Conservation of momentum states that the total momentum of a closed system before a collision is equal to the total momentum after the collision, provided no external forces act on the system.

**Distinguish between an elastic and inelastic collision.****Answer**: In an elastic collision, both momentum and kinetic energy are conserved. Objects “bounce” off each other. In an inelastic collision, momentum is conserved but kinetic energy is not. Objects might stick together or deform after the collision.

**How is it possible for a small force acting over a long time to produce the same change in momentum as a large force acting over a short time?****Answer**: Because impulse is the product of force and time, a small force acting over a longer duration can yield the same impulse (and thus the same change in momentum) as a larger force acting over a shorter duration.

**If a car crashes into a wall and comes to a stop, is momentum conserved?****Answer**: For the car alone, momentum is not conserved because it comes to a stop. However, in the broader system (including the Earth and the wall), momentum is conserved. The momentum imparted to the car is imparted in an equal and opposite manner to the Earth and wall, but due to the vast difference in mass, the Earth’s change in velocity is imperceptibly small.

**Why do airbags in cars help reduce injuries during collisions?****Answer**: Airbags increase the time over which a person’s momentum is changed as they come to a stop, reducing the average force experienced during the collision. This decrease in force helps to minimize injuries.

**If two objects with the same mass have opposite velocities of equal magnitude and collide head-on, what will be their combined velocity after the collision?****Answer**: Assuming an elastic collision and no external forces, the objects will bounce back with the same speed but in the opposite direction. If the collision is perfectly inelastic, they will stick together and come to a stop since their momenta will cancel each other out.

**Why do bouncing balls eventually come to a stop even if they are in a vacuum (with no air resistance)?****Answer**: While a vacuum eliminates air resistance, it doesn’t prevent the ball from undergoing inelastic collisions with the ground. Each time the ball bounces, some kinetic energy is converted to other forms (like sound or deformation energy), causing the ball to bounce lower with each successive bounce until it stops.

**How can momentum be “hidden” in systems, such as rotating objects?**

**Answer**: While linear momentum pertains to the straight-line motion of objects, rotating objects have angular momentum. An object can have zero linear momentum but significant angular momentum if it’s rotating. For example, a spinning top at rest on a table has no linear momentum but has angular momentum due to its spin.