4.2 Impulse and Momentum

Linear Momentum🥚

In this unit, we'll only be talking about linear momentum, but look forward to angular momentum in the next unit! Conceptually, its quite hard to describe without using quantities, but essentially, it is a measurement of mass in motion. The most basic equation for linear momentum is:

p = mv

Momentum Fast (haha) Facts:

• It's a vector
• Units are kg*m/s
• It is NOT the same as kinetic energy- Kinetic energy is a scalar and describes a different relationship Let's try to derive Newton's Second Law using momentum!

Now that looks pretty familiar, doesn't it?

Let's take a closer look at one part of this derivation.

This change in momentum actually has a special name, Impulse! (Or as some like to call, Jimpulse)

Or the calculus version:

We tend to use calculus when the force or the mass is variable, like with a rocket!

Impulse

Impulse is a vector because force has a direction, and the units are the same as momentum(N*s). 

Here are some key things to know about impulse:

• Impulse is the product of force and time, represented mathematically as J = F*Δt.
• Impulse has the same effect on an object as a constant force applied over a certain period of time.
• Impulse can be used to analyze the effect of forces on moving objects, such as during collisions.
• Impulse can be used to find the final velocity of an object after a force has been applied for a given period of time.
• Impulse can also be used to find the force required to stop an object or change its velocity by a certain amount in a specific period of time.
• Impulse can be used in the context of both elastic and inelastic collisions, where inelastic collisions are defined as collisions where kinetic energy is not conserved and elastic collisions are defined as collisions where kinetic energy is conserved.
• Impulse can be used to analyze the motion of an object in a resistive medium, such as air resistance or friction.
• The impulse-momentum theorem states that the impulse applied to an object is equal to the change in momentum of the object, mathematically represented as J = Δp.
• The conservation of momentum states that the total momentum of a closed system remains constant, unless acted upon by an external force.
• In a collision, the total impulse experienced by the two colliding objects must be equal and opposite, this is also known as Newton's third law.

Additionally, AP loves to ask questions about impulse in relation to graphs!

Impulse is the area under the curve (check out the integral!) of a Force vs. Time graph.

Taken from Physics Forums

The curves won't always look as pretty as they did in AP Physics 1 because we have the power of calculus now!

⚠️ So..what does impulse really mean?

Have you ever tried an egg drop experiment? One where you tried to shield a poor egg from cracking as soon as it hit the ground?

Think back to which experiments worked best...theoretically the ones that work best should be the ones that increase the amount of time of impact!

Center of Mass Velocity

The center of mass of a system of objects is a point that represents the average position of all the objects in the system. The velocity of the center of mass of a system of objects is the rate of change of the position of the center of mass with respect to time. 

Here are some key things to remember when solving a problem asking for the velocity of the center of mass:

• The momentum of an object is equal to its mass multiplied by its velocity, represented mathematically as p = m*v.
• The total momentum of a system of objects is equal to the sum of the momenta of all the individual objects in the system.
• The center of mass of a system of objects will have a velocity that is directly proportional to the total momentum of the system and inversely proportional to the total mass of the system.
• In other words, if the total momentum of a system of objects increases, the velocity of the center of mass will also increase, and if the total mass of a system of objects increases, the velocity of the center of mass will decrease.
• For a closed system, if no external forces are acting on it, the total momentum of the system will remain constant, which means that the velocity of the center of mass will also remain constant.
• In the context of collisions, the total momentum before and after the collision must be the same, assuming no external forces are acting on the system. This means that the velocity of the center of mass will change accordingly.

Practice Questions

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?(Taken from Lumen Learning)

Answer: To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.

Now the magnitude of the net external force can determined by using 

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Answer:

Use the Impulse formula on your formula chart, and don't forget to include your bounds/limits of integration!

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Taken from College Board

Answer:

Again, use the formula on your chart and integrate with bounds. Take note that there is an alternate form of resolving this with another impulse formula