A Review of Some Newtonian Mechanics Gernot Laicher

A Review of Some Newtonian Mechanics Gernot Laicher REFUGES Summer Bridge Course 2015

Position, Velocity, Acceleration

Mass, Force

Momentum m

Sir Isaac Newton Godfrey Kneller’s 1689 portrait of Isaac Newton (age 46) (Source: http: //en. wikipedia. org/wiki/Isaac_Newton)

Newton's First Law : An object at rest tends to stay at rest and an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. Newton's Second Law: An applied force on an object equals the rate of change of its momentum with time. Newton's Third Law: For every action there is an equal and opposite reaction.

Newton’s Second Law

For a non-changing mass:

Rewriting Newton’s Second Law Impulse = Change in Momentum

Example: A constant force of 5 N applied for 0. 5 seconds on a mass of 0. 2 kg, which is initially at rest.

In general, the force may depend on time Impulse = “Area” under the F(t) curve (Integrate F(t) over time to get the impulse)

A Two-body Collision Before the collision During the collision (Newton’s third law)

During the Collision Impulse on m 1 Impulse on m 2 = - Impulse on m 1

Impulse on m 2 = - Impulse on m 1 Change of momentum of m 2 = - Change of momentum of m 1 Total momentum after collision = Total momentum before collision The total momentum of the system is “conserved” during the collision. (This works as long as there are no external forces acting on the system)

After the collision For a 1 -dimensional collision we can replace the vector with + or – signs to indicate the direction. Note: u 1, u 2, v 1, v 2 may be positive or negative, depending on direction and depending on your choice of coordinate system.

Given: m 1 , m 2 , v 1 , v 2 What can we find out about the final velocities u 1 and u 2 ?

Given: m 1 , m 2 , v 1 , v 2 What can we find out about the final velocities u 1 and u 2 ? Answer: In general there an infinite number of possible solutions (combinations of the final velocities u 1 and u 2 that fulfill the conservation of momentum requirement). Which of those solutions really happens depends on the exact nature of the collision.

Excel Program – Practicing the Solver 1) Solving a simple math problem (2 x + 6 y =14) infinitely many solutions 2) Adding a constraint (a second equation) (x+y=3) to get a single solution. 3) Solving a quadratic equation (4 x 2 + 2 x =20) how many solutions do you get? 4) Pick different initial conditions for x to find all the solutions. 5) Find a solution to the following set of equations: 3 x 2 + 4 y + 12 z = 29 4 x - y 2 – z = 12 x + 2 y - 5 z = 5

Excel Program – Using the Solver to determine 1 -dimensional collision outcome 1) Write an Excel program that has the following input fields: m 1 , m 2 , v 1 , v 2 , u 1 , u 2. 2) Create (and label) fields that calculate p 1 initial , p 2 initial , p 1 final , p 2 final. 3) Create (and label) fields that calculate ptotal initial , ptotal final. 4) Create (and label) fields that calculate ptotal final - ptotal initial. 5) Fill in these values: m 1=1 (kg), m 2=1 (kg), v 1=1 (m/s), v 2=-1 (m/s), u 1=some value (m/s), u 2=some value (m/s). 6) Use a solver that changes u 1 and makes ptotal final - ptotal initial = 0. How many possible solutions can you find?

Mechanical Energy of Two Mass System Before the collision After the collision

Totally Elastic Collision: (no mechanical energy is lost)

Excel Program – Using a Solver Add the following to your Excel “Collision-Solver”: Create fields that calculate Etotal initial , Etotal final - Etotal initial , % Energy change. Now solve your collision problem again. This time, use the constraint Etotal final - Etotal initial = 0 Find all possible solutions and interpret their meaning.

Totally Inelastic Collision: (The two masses stick together after the collision and move with the same velocity) Task: Determine a mathematical formula for the final velocity after a totally inelastic collision as a function of the masses and the initial velocities of two colliding objects.

Totally Inelastic Collision: (The two masses stick together after the collision and move with the same velocity)

Is there Mechanical Energy Conservation in Totally Inelastic Collisions?

Excel Program – Using a Solver Name the Excel worksheet tab for the previously created elastic collision “Elastic Collision”. Make a copy of this tab and rename it “Inelastic Collision” Modify the “Inelastic Collision” tab so that it calculates “u 1” from the initial conditions (masses, initial velocities) according to the formula for u on the previous page. Then you can make u 2=u 1. So now both u 1 and u 2 will always have the same value.

Use a lab notebook to record your findings and investigate several scenarios of collisions For both elastic and inelastic collisions you should find solutions for these cases. Imagine each scenario and describe what is happening. 1) m 1=5 kg, m 2=5 kg, v 1=15 m/s, v 2=0 (one mass initially at rest) 2) m 1=5 kg, m 2=5 kg, v 1=15 m/s, v 2=-15 m/s (one mass initially at rest) (head on collision). 3) m 1=5000 kg, m 2=0. 01 kg, v 1=15 m/s, v 2=0 (very unequal masses, lighter mass at rest). 4) m 1=5000 kg, m 2=0. 01 kg, v 1=0, v 2=-15 m/s (very unequal masses, heavier mass at rest). 5) m 1=5000 kg, m 2=0. 01 kg, v 1= 15 m/s, v 2=-15 m/s(head on collision)

Quick Review: Vectors and Scalars • Vectors: Magnitude and Direction (e. g. , Force, Displacement, Velocity, Acceleration) • Scalars: Magnitude only (e. g. , Temperature, Altitude, Age)

Adding Vectors Two forces are acting on a mass m: How do we find the total force ?

Subtracting Vectors An object changes it’s velocity. What is it’s change in velocity?

(Note: F 2 y had a negative value)