1. “But, if this is true, and if a large stone moves with a speed of, say, eight while a smaller one moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter, an effect which is contrary to your supposition. Thus you see how, from your supposition that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.” 4 8 < 8 ? > 8 ? v v v v Galileo’s Argument Regarding Freely Falling Bodies

2. Newton’s Laws of Motion I.Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. II.The change of motion is proportional to the motive force impressed and is made in the direction of the line in which that force is impressed. III.To every action there is always imposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and opposite.

3. Aristotelian View  The natural state of objects on earth is at rest with respect to the earth.  The natural state of heavenly bodies is in circular motions. In some sense, Newtonian mechanics unified the physics of the heavens and earth.  The development of modern science is, to some extent, the concession that experiment dictates “truth”, and that the natural world can not be apprehended by pure thought alone.  Scientific truth amounts to that explanation which has predictive value, is the most broadly applicable, and is the most simple.

4. Newton’s 1 st Law Inertia Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.  Mass is a measure of inertia. Inertia is the resistance to change in an object’s momentum.  Momentum is always measured with respect to some fixed reference point.  A reference frame is a coordinate system fixed to some reference point.  An inertial reference frame is a non- accelerating reference frame, i.e., one in which the Newton’s laws are valid.

5. The Fly in the Inertial Reference Frame Ointment Is this an accelerating reference frame or is there simply an external force acting?

6. Definition 1.1.1: A differential equation is an equation involving one or more derivatives of an unknown function. Newton’s Second Law is a Differential Eqn. Ordinary Differential Eqn. (ODE) ODE Partial Differential Eqn., (PDE)

7. Definition 1.1.2: The order of the highest derivative occurring in a differential equation is called the order of the differential equation. Has order 1 Has order 2. Has order 4. Partial Differential Equation. Has order 1

8. Definition 1.1.3: A differential equation that can be written in the form where and F are functions of x only, is called a linear differential equation of order 1. Such an equation is linear in y, y’, etc.

9. nonlinear linear Examples of Linear and Nonlinear Equations

10. General Form For 1 st and 2 nd Order Linear Differential Equations General Form For 1 st Order Linear ODE General Form For 2 nd Order Linear ODE

11. Solutions of Ordinary Differential Equations Definition 1.1.4: A solution of an nth-order differential equation on an interval I is any function of the form x = x(t) that is (at least) n times differentiable on I and that satisfies the differential equation identically for all t in I. Show that: Where c 1 and c 2 are constants, is a solution of the linear differential equation: Furthermore, x(t) is twice differentiable For all real t. Differentiate: Differentiate again:

12. Definition 1.1.5: A solution to an n th - order ordinary differential equation on an interval I is called the general solution on I if it satisfies the following conditions: 1.The solution contains n constants, c 1, c 2,…, c n. 2.All solutions of the differential equation can be obtained by assigning appropriate values to the constants. General Solution Remark: Not all differential equations have a general solution. For example, consider: The only solution to this differential eqn. Is y(x)=1. Hence the differential equation doesn’t have a solution containing an arbitrary constant.

13. Example Find the general solution of the differential equation: Solution: Write the equation as Integrate Integrate again to find the general solution

14. Note A solution of a differential equation is called a particular solution if it does not contain any arbitrary constants. One way in which particular solutions arise is by assigning specific values to the arbitrary constants occurring in the general solution of a differential equation. For example, from the previous example: General solution A particular solution

15. Falling ball with no air resistance: mgmg Positive y direction. Draw picture Make a free body diagram Choose a coordinate system Write down F=ma Decompose into component equations Consider the following:

16. Integrate: Integrate again: The above is the general solution. To determine the motion of a particular object, we need to know: These auxiliary conditions are called initial conditions since they are imposed at the same value, to, of the independent variable.

17. The corresponding problem: Solve Subject to Is referred to as an initial value problem. Definition 1.1.6: An n th order differential equation together with n auxiliary conditions imposed at the same value of the independent variable is called an initial value problem. Initial Value Problems