Lec 02: Introduction to Kinematics | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Scientific concepts & nature/physical phenomena presented

1) 1D motion, position vs time, and sign conventions

• Objects move along a straight line (1D motion).
• An (x(t)) diagram (position as a function of time) represents the “road” of motion.
• The direction where (x) increases is arbitrary (choice of sign convention).
  • Changing where (x=0) is placed does not affect average velocity values.
  • Reversing the direction of increasing (x) flips the signs of velocity and acceleration.

2) Average velocity (over a finite time interval)

Average velocity between times (t_1) and (t_2) is: [ \bar{v}=\frac{x(t_2)-x(t_1)}{t_2-t_1} ] Key ideas from the examples:

• Nonzero average velocity occurs when start/end positions differ.
• Average velocity can be zero even if the object moves, e.g., when it returns to the same position.

3) Average speed vs average velocity (speed is sign-insensitive)

• Average velocity depends on displacement (positive and negative can cancel).

• Average speed depends on total distance traveled (never cancels).

  • [ \text{Average speed}=\frac{\text{total distance}}{\text{total time}} ]

• Example: if an object returns to its starting position:

  • average velocity = 0
  • average speed > 0

4) Instantaneous velocity and the derivative

Instantaneous velocity is defined by the limit: [ v(t)=\lim_{\Delta t\to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t} ] A core equation emphasized: [ v=\frac{dx}{dt} ] Connection to the slope of the (x)–(t) curve:

• At turning points (object reverses direction), velocity = 0.
• The sign of velocity depends on whether the slope is positive or negative relative to the chosen “(x) increasing” direction.

5) Instantaneous speed

• Speed = magnitude of velocity (always nonnegative).
• Velocity can be positive or negative, but speed increases with larger (|v|).

6) Average acceleration

Average acceleration between times (t_1) and (t_2): [ \bar{a}=\frac{v(t_2)-v(t_1)}{t_2-t_1}=\frac{\Delta v}{\Delta t} ]

• Sign depends on conventions (how (x) is defined as increasing).
• Units: (\text{m/s}^2).

7) Impact examples: bounces, tomatoes/eggs, and “what causes breaking”

• Tennis ball bounce
  • Velocity reverses direction over a short impact time.
  • Produces large acceleration (its magnitude depends on (\Delta v) and the impact duration).
• Tomato/egg thrown and not bouncing back
  • Smaller (\Delta v) and longer impact time.
  • Leads to smaller average acceleration magnitude than in a bounce.
• Key point:
  • “What breaks” depends on acceleration magnitude and impact time, not on sign conventions.

8) Instantaneous acceleration and second derivatives

Instantaneous acceleration: [ a(t)=\lim_{\Delta t\to 0}\frac{v(t+\Delta t)-v(t)}{\Delta t}=\frac{dv}{dt} ] Also emphasized: [ a=\frac{d^2x}{dt^2} ] Qualitative method from a velocity plot:

• If velocity is constant → acceleration = 0
• If velocity increases → acceleration > 0
• If velocity decreases → acceleration < 0
• Requires attention to how the slope of (v) vs. (t) changes.

9) Worked kinematics example: (x(t)=8-6t+t^2)

Given: [ x(t)=8-6t+t^2 ] Derived: [ v(t)=\frac{dx}{dt}=-6+2t ] [ a(t)=\frac{dv}{dt}=2 ] So acceleration is constant.

Solved values:

• Position zeros: (t=2) and (t=4)
• Velocity zero: (t=3)
• Position at (t=3): (x=-1)

Interpretation:

• Negative initial velocity slows to a stop, then reverses direction.
• Positive constant acceleration increases velocity over time.

10) Constant acceleration kinematics (general form)

For constant acceleration in 1D, position is quadratic in time: [ x=C_1+C_2 t+C_3 t^2 ] Interpretation:

• (C_1): position at (t=0)
• (C_2): velocity at (t=0)
• Acceleration: (a=2C_3) (constant)

11) Gravity as constant acceleration and the gravitational constant (g)

Under idealized conditions (vacuum, no air drag), gravitational acceleration is:

• Independent of mass, composition, and shape.
• Typical value near Boston: [ g \approx 9.80\ \text{m/s}^2 ]

Vertical motion formulas: [ x=x_0+v_0 t+\tfrac{1}{2} g t^2 ] [ v=v_0+gt ] [ a=g ]

Conceptual note:

The independence from many properties is accepted based on experimental support (not derived from first principles within the course context).

12) Experiment: timing accuracy, stroboscopic measurement, and free fall

Two demonstration activities are described:

A) Bullet speed measurement using two wires and uncertainties

Setup

• Two wires (I and II) separated by distance (D).
• Bullet breaks wire I → timer starts.
• Bullet breaks wire II → timer stops.

Uncertainty accounting

• Distance uncertainty: about (\pm 0.5) cm around (148.5) cm.
• Timing uncertainty: about (\pm 0.1) ms.

Goal

• Target about 2% speed accuracy, requiring appropriate timing resolution based on an estimate of bullet speed (order of 200–300 m/s, comparable to sound speed (\sim 340) m/s).

Reported result (with propagated uncertainty)

• Measured time: (\sim 5.8) ms
• Speed: (\sim 256) m/s with (\sim 4) m/s uncertainty (≈1.7% timing-dominated error).

B) Apple free-fall stroboscope + phototiming “alertness test”

First test

• Height: about 3 m (expected fall time (\sim 0.8) s).
• Strobe flashes at 2 Hz (twice per second) to observe two positions.
• Since strobe and release may not be synchronized, observers must infer where the apple is at each flash.
• A photograph is used to determine the exact positions.

Second test

• Strobe frequency increased to 10 Hz for about 0.8 s of falling.
• Students count the number of observed “balls”/positions.
• The count depends on how the strobe phase intersects the fall.
• Demonstration shows observers can miss positions; an example includes approximately 7 observed positions for a “ten hertz” scenario (with the exact number depending on timing phase).

Researchers or sources featured (at end)

• Walter Lewin (instructor; narrator of the lecture)
• Sherlock Holmes (fictional character referenced in an example)
• Dr. Watson (fictional character referenced in an example)