Displacement is the net change in position. It is described by Δx. For example, if a particle begins at +2 on an axis and then after 60 seconds is at +12, then the displacement is 10. The displacement will still be 10 even if the particle travelled to -4, +60, and then -7 in that time. As long as it winds up at +12, the displacement will be 10.

The total distance, however, is the total distance covered. In the case above, the total distance would be 153.

Use the pythagorean theorem sometimes. Give examples.

Average Velocity is displacement/time.

Speed is distance/time.

Instantaneous velocity is the velocity of partice at any instant of time.

The instantaneous velocity is v_x = lim_ΔT→0 ΔX/ΔT

This limit is called the derivate of x with respect to t. The instantaneous velocity can be positive, negative, or zero. When the slope of the position-time graph is positive, vx is positive.

The instantaneous speed of a particle is defined as the magnitude of the instantaneous velocity vector. Hence, by definition, speed can enver be negative.

A model is a simplified substitute for the real problem, allowing us to solve the problem in a relatively simple way.

• Analysis models are previously solved problems that describe either (a) the behavior of some physical entity or (b) the interaction between that entity and the environment.
• Geometric models form geometric constructions of real situations. We then analyze the geometric construction. These are commonly used in trigonometry.
• Simplification models ignore insignificant details.
• Structural models make it easier to understand concepts which are relatively abstract. For example, thinking of an election as a cloud of electron density is a structural model of an electroon.

Thinking of problems from different perspectives will oftentimes help you figure things out.

• Mental representations occur when you imagine a scene based on a description in a word problem. You envision a sequence of events and can predict what will occur in the future.
• Pictorial representations occur when you draw your mental representaition. The drawing describes what you would see if you were observing the situation in the problem.
• Simplified pictorial representations are pictorial representations to which a simplification model has been applied.
• Graphic representations are graphs which represent situations in problems

A particle is accelerating when it’s velocity changes with time. Acceleration measures how rapidly velocity is changing. The average acceleration, a_x,avg, of the particle in the time interval ΔT = t_ƒ – t_i is: Δv_x / Δt.

The average acceleration may be different for different time intervals. Instantaneous acceleration is limit of average acceleration as Δt → 0:

a_x = lim_Δt→0 Δv_x/Δt

Please note that force is proportional to acceleration:

F ∝ a

A freely falling object moves freely under the influence of gravity alone. When air resistance is ignored (a vaccuum is a simplification model) then the object is in free fall. Magnitude of free-fall acceleration is denoted by g. At the surface of Earth, g is approximatel 9.80 m/s², or 980 cm/s², or 32 ft/s². It can be assumed that the vector g is directed toward the center of Earth.

Instantaneous position: x(t)
Instantaneous velocity: v(t) = dx/dt
Instantaneous acceleration: a(t) = dv/dt = d²x/dt²

A straight line curve has constant velocity, meaning it lacks acceleration. There are several important formulas:

• Sample Problem

  If you throw up a basketball, how long does it take to hit the ground? Gravity is 9.8m/s².

  x_i = +1m
  x_f = 0m
  v_i = +5m/s
  a= = -9.8 m/s² ≈ -10m/s²

  ½gt² – v_it + (x_f – x_i) = 0

  t_{1 or 2} = (-b ± √(b² – 4ac) )/(2a) = v_i &#177 √(v_i² – 2g(x_f – x_i) )/g

  DET: v_i² – 4½g(x_f – v_i) = 25m½/s½ – 20m/s½ (0 -1) = 45m½s½

  t_{1 or 2} = (5m/s ± √(45m/s) )/10m/s² = 1.17 s or -0.17 s

  v_f – v_i = -g(v_i ± √(v_i² – i₀) )/g

  v_f – v_i = -v_i &#177 -√(v_i² – 2g(x_f – x_i))

  v_f = √(v_i² – 2g(x_f – x_i) ⇒ v_f² – v_i² = -2g (x_f – x_i)

Important Note: if you throw up a ball with a certain v_i, and you thrown down a ball with the same v_i, they will both have the same velocity when they hit the ground.

In uniform circular motion, a particle moves in a circle at constant speed. However, the acceleration is constantly changing. This is because acceleration is a change in velocity, the velocity vector is changing direction. The acceleration vector is always perpindicular to the path and always points toward the center of te circle. This is centripetal acceleration, and its magnitude is defined by the equation below. r represents the radius of the circle. The subscript on the acceleration symbol means that the acceleration is centripetal.

a_c = v²/r