PHYS101: Introduction to Mechanics

7a. Define linear momentum

• Define linear momentum and the equation we use to calculate it.
• Why is linear momentum a vector?
• What are the units for linear momentum?

We define linear momentum as the product of an object's mass and velocity. It can be written as $p = mv$, where $p$ is linear momentum, $m$ is mass, and $v$ is velocity.

Linear momentum is a vector quantity because velocity is a vector quantity, and the linear momentum will have the same direction as the velocity. The units for linear momentum are kgm/s.

7b. Use Newton's Second Law in terms of momentum

• Describe how we can reframe Newton's Second Law of Motion in terms of momentum.
• Use the equations from Newton's Second Law of Motion to calculate force.

We reviewed in Unit 4 learning outcome 4b that we can write Newton's Second Law of Motion as $F= ma$. While this is the most common way to write and use this law, it was not how Newton originally wrote it. Newton wrote this law in terms of momentum rather than force and acceleration:

$F_\mathrm{net} = \frac{\Delta p}{\Delta t}$

This shows that the net force equals the change in momentum divided by the change in time. This equation certainly appears different from the familiar $F=ma$ we used in Unit 4.

Review the derivation of how $F=ma$ can be obtained from the Second Law in terms of momentum in equations 8.9, 8.10, 8.11, and 8.12 in Linear Momentum and Force.

Review a worked example of using Newton's Second Law in terms of momentum in Example 8.2: Calculating Force: Venus Williams' Racquet. This problem calculates the force applied to a tennis ball: there is a change of velocity of the ball but no change in mass, so pay special attention to how change in momentum is calculated in equation 8.14.

7c. Describe the relationship between impulse and momentum

• Define impulse.
• What is the assumption we make when calculating impulse?

We define impulse as change in momentum. Using Newton's Second Law of Motion, we can write this as:

$\Delta p=F_\mathrm{net} \Delta t$

When we calculate impulse, we assume the net force is constant during the time we are interested in. In reality, force is rarely constant. For example, in Example 8.2, we assumed the force on the tennis ball was constant over time. In reality, the force on the tennis ball probably changed from the beginning to the end of the swing of the tennis racquet. However, the change in force was probably not significant, and we assume it is constant to make our calculations easier. 

7d. Define elastic, inelastic, and totally inelastic collisions

•  Define elastic, inelastic, and totally inelastic collisions. 
•  Give examples of each type of collision.

When two or more objects physically interact, we say the objects collide or have a collision. There are three types of collisions we can consider when solving physics problems, which are all based on energy transfer in the collisions. 

By definition, an elastic collision is a collision in which the internal kinetic energy is conserved in the interaction. Internal kinetic energy is the sum of the kinetic energy of all of the objects colliding. So, in an elastic collision, all the kinetic energy remains kinetic energy. That is, no kinetic energy is converted to heat, friction, or other types of energy. In reality, no collisions are perfectly elastic because some kinetic energy is always "lost" by being converted to other forms of energy. A close example of an elastic collision is if two balls collide on a smooth icy surface. Because the ice has almost no friction, little kinetic energy would be lost to friction.

Review a diagram of two metal boxes interacting in an elastic collision on an ice surface in Figure 8.6.

Unlike an elastic collision, an inelastic collision is a collision in which the internal kinetic energy is not conserved. In inelastic collisions, some kinetic energy of the colliding objects is lost to friction, heat, or even work. Inelastic collisions are what we mostly observe in the real world.

Review a good example of an inelastic collision in Figure 8.9. In this example, a hockey goalie stops a puck in the net. Although the ice surface is essentially frictionless, some kinetic energy of the puck is converted to heat and sound as the goalie stops it. A totally inelastic collision (also called perfectly inelastic collision) is an inelastic collision in which the objects "stick together" upon colliding.

Review an example of two blocks experiencing a totally inelastic collision in Figure 8.8. 

7e. Use conservation of momentum to solve collision problems

• Calculate velocities of objects following an elastic collision.
• Calculate velocities of objects and energy lost following an inelastic collision.

When solving problems for elastic collisions, it is important to remember that the internal kinetic energy is conserved. Therefore, the total kinetic energy at the start of the collision must equal the total kinetic energy at the end of the collision.

We can write this as: $\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}m_1 v_1^{'2} + \frac{1}{2}m_2 v_2^{'2}$

Moreover, we know that momentum must be conserved in the collision. Therefore, the total momentum at the start of the collision must equal the total momentum at the end of the collision. That is, for two objects (object one and two) colliding, we can write: $\frac{1}{2}m_1 v_1 + \frac{1}{2}m_2 v_2 = \frac{1}{2}m_1 v_1^{'} + \frac{1}{2}m_2 v_2^{'}$. Using conservation of momentum, we can usually set up these problems so we only have to solve for one unknown.

Review a worked example of this type of problem in Example 8.4: Calculating Velocities Following an Elastic Collision. In this example, one of the objects is initially at rest (its velocity equals zero), so it does not have an initial momentum. This lets us simplify the conservation of energy momentum equations. Then, by using the equations for conservation of energy and momentum, we can solve for final velocity after collision.

We can also solve for final velocities after inelastic collisions. In these problems, it is important to remember that kinetic energy is not conserved but momentum is conserved. Example 8.5 (a): Calculating Velocity and Change in Kinetic Energy: Inelastic Collision of a Puck and a Goalie shows a worked example of this type of problem. Here, the conservation of momentum equation is used to determine the final velocity of the object (the hockey goalie) in an inelastic collision.

In inelastic collisions, some kinetic energy is converted to other forms of energy. The energy difference before and after collision can be calculated to determine how much kinetic energy was lost. Example 8.5 (b): Calculating Velocity and Change in Kinetic Energy: Inelastic Collision of a Puck and a Goalie shows a worked example of calculating the energy lost in this inelastic collision. Here, the total kinetic energy in the system is calculated before and after collision based on the mass and velocities of the objects. The difference in kinetic energy shows how much kinetic energy was converted to other forms of energy during the collision.

Example 8.6: Calculating Final Velocity and Energy Release: Two Carts Collide is a similar worked example.

7f. Demonstrate the physics behind rocket propulsion

• Which law of motion governs rocket propulsion? Why?
• Give some examples of rocket propulsion in nature and technology.
• What makes a rocket accelerate?

Rocket propulsion is directly tied to Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction (review learning outcome 4d). In rocket propulsion, matter is removed from a system with a strong force. This causes an equal force in the opposite direction on the system. An example is the recoil of a gun. When a gun shoots, the bullet is removed from the system (the gun) with a strong force. Because of Newton's Third Law, the gun itself experiences recoil and pushes back away from the bullet's path. Another example is the movement of squid. To move, squid push water out of their bodies with a strong force. This causes the squid to move in the opposite direction of the water being pushed out of the squid's body.

Figure 8.13 shows a diagram of a rocket going straight up. When the rocket launches, it expels a large mass of hot gas (from the fuel) down toward the earth. As the gas is expelled, the rocket's velocity increases because its overall mass decreases. The acceleration of the rocket is proportional to its change in mass as it burns its fuel. The faster the fuel is burned, the greater the acceleration of the rocket.

Unit 7 Vocabulary

• Elastic collision
• Impulse 
• Inelastic collision
• Internal kinetic energy
• Linear momentum
• Momentum
• Newton's Second Law of Motion (in terms of momentum)
• Rocket propulsion
• Totally inelastic collision