General Physics 1: Motion Descriptors and Uniformly Accelerated Linear Motion Notes

Introduction to Motion and Motion Descriptors

Motion is a fundamental concept in physics, specifically within the study of kinematics.

• Definition of Motion: Motion is defined as the change of position of an object in a specific span of time relative to an observer. To identify if an object is moving, one must observe its position compared to a reference point over a duration of time.
• Essential Components of Motion:
  • Position: The specific location of an object. To measure motion, we track the initial position ($x_0$) and the final position ($x_f$).
  • Time: The duration during which the change in position occurs ($t$).

Detailed Motion Descriptors

There are six primary descriptors used to quantify and qualify the motion of an object.

Time

• Definition: A quantity that describes when an event took place. It is a necessary parameter to observe changes in a specific space.
• Symbol: $t$
• SI Unit: seconds ($s$)

Distance and Displacement

Distance and displacement both measure how far an object has moved, but they differ in their nature as scalar or vector quantities.

• Distance:

  • Definition: Describes the total length traveled by an object in motion. It accounts for the entire path taken.
  • Type of Quantity: Scalar (magnitude only).
  • Symbol: $x$, $y$, $z$, or $d$
  • SI Unit: meters ($m$)
  • Formula: $d = \sum |x_n|$ (Total length travelled).

• Displacement:

  • Definition: The length and direction of the straight line that connects the initial position to the final position. It describes how far an object is from its starting point.
  • Type of Quantity: Vector (magnitude and direction).
  • Symbol: $\Delta x$, $\Delta y$, $\Delta z$, or $\vec{d}$
  • SI Unit: meters ($m$)
  • Formula: $\Delta x = x_f - x_i$

• Relationship: Distance is always greater than or equal to displacement ($d \ge |\Delta x|$). They are equal only if the object travels in a single straight line without reversing direction.

Speed and Velocity

These quantities combine the concepts of space (displacement/distance) and time.

• Speed:

  • Definition: The rate of change in position; how fast an object is changing its position within a span of time.
  • Type of Quantity: Scalar.
  • Symbol: $v$ or $s$
  • SI Unit: meters per second ($m/s$ or $m\,s^{-1}$
  • Formula: $v = \frac{d}{t}$

• Velocity:

  • Definition: The rate of change in position with respect to a reference point and direction.
  • Type of Quantity: Vector.
  • Symbol: $\vec{v}$
  • SI Unit: meters per second ($m/s$)
  • Formula: $\vec{v} = \frac{\Delta x}{t}$

Acceleration

• Definition: The rate of change in the velocity of an object.
• Criteria for Acceleration: An object is accelerating if:
  1. The magnitude of the velocity changes (speeding up or slowing down).
  2. The direction of motion changes.
  3. Both the magnitude and the direction of the velocity change.
• Type of Quantity: Vector.
• SI Unit: meters per second squared ($m/s^2$
• Formula: $a = \frac{v_f - v_i}{t}$

Signs of Acceleration

• Positive Acceleration: The acceleration ($a$) and velocity ($v$) are in the same direction. The object is speeding up.
• Negative Acceleration (Deceleration): The acceleration ($a$) and velocity ($v$) are in opposite directions. The object is slowing down.
• Zero Acceleration: The object is either at rest or traveling at a constant velocity.

Uniform Acceleration and Kinematic Equations

Uniform acceleration occurs when the velocity of an object changes at a fixed rate throughout the motion.

Constant Velocity vs. Constant Acceleration

• Constant Velocity: The object has a constant magnitude and direction. It covers equal displacements in equal time intervals.
• Constant Acceleration: The velocity is not constant, but the rate of change of velocity is constant.

The Four Kinematic Equations (Constant Acceleration Equations - CAE)

These equations are used to solve for unknown variables in one-dimensional uniformly accelerated motion. Each equation is independent of one specific motion descriptor.

1. Displacement-Independent Equation:     $v = v_0 + at$

2. Acceleration-Independent Equation:     $\Delta x = \frac{v + v_0}{2} t$

3. Final Velocity-Independent Equation:     $\Delta x = v_0 t + \frac{1}{2} a t^2$

4. Time-Independent Equation:     $v^2 = v_0^2 + 2 a \Delta x$

Where:

• $v$ = final velocity
• $v_0$ = initial velocity
• $a$ = acceleration
• $t$ = elapsed time
• $\Delta x$ = displacement ($x - x_0$

Problem-Solving Strategies

To convert verbal descriptions of motion into mathematical equations, follow these steps:

1. Read the problem carefully.
2. Identify Given Values: List all known variables ($v, v_0, a, t, \Delta x$).
3. Identify the Unknown: Write what variable is being asked for.
4. Select the Equation: Choose the kinematic equation that contains the given values and the unknown variable.
5. Watch for Implicit Givens: Phrases like "starts from rest" imply $v_0 = 0\,m/s$. Phrases like "comes to a stop" imply $v = 0\,m/s$.
6. Unit Consistency: Ensure all units match (e.g., all in meters and seconds) before calculation.

Examples and Applications

Example 1: Calculating Acceleration

A car uniformly accelerates from rest to reach a maximum velocity of $16.67\,m/s$ in $10\,s$.

• Given: $v_0 = 0$, $v = 16.67\,m/s$, $t = 10\,s$
• Formula: $a = \frac{v - v_0}{t}$
• Solution: $a = \frac{16.67\,m/s - 0\,m/s}{10\,s} = 1.667\,m/s^2$
• (Note: The transcript result says $2\,m/s^2$ likely as a rounded instructional placeholder, but the calculated value based on given numbers is $1.667\,m/s^2$).

Example 2: Calculating Distance (Acceleration-Independent)

A ball rolled at a speed of $20\,m/s$ and changed its speed to $3\,m/s$ in $8\,s$. What is the distance?

• Given: $v_0 = 20\,m/s$, $v = 3\,m/s$, $t = 8\,s$
• Formula: $\Delta x = \frac{v + v_0}{2} t$
• Solution: $\Delta x = \frac{3\,m/s + 20\,m/s}{2} \times 8\,s = 11.5\,m/s \times 8\,s = 92\,m$

Example 3: Constant Acceleration involving Maria

Maria rides her bicycle at an initial speed of $2\,m/s$. She accelerates at $1\,m/s^2$ for $4\,s$. How far did she travel?

• Given: $v_0 = 2\,m/s$, $a = 1\,m/s^2$, $t = 4\,s$
• Formula: $\Delta x = v_0 t + \frac{1}{2} a t^2$
• Solution: $\Delta x = (2\,m/s)(4\,s) + \frac{1}{2}(1\,m/s^2)(4\,s)^2 = 8\,m + 8\,m = 16\,m$

Example 4: The Boar Problem (Two Unknowns)

A boar runs a distance of $70\,m$ in $6\,s$. It reaches the second point with a velocity of $15\,m/s$. Calculate the initial velocity and acceleration.

• Given: $\Delta x = 70\,m$, $t = 6\,s$, $v = 15\,m/s$
• Step 1: Solve for $v_0$ using $\Delta x = \frac{v + v_0}{2} t$
  • $70 = \frac{15 + v_0}{2} \times 6$
  • $70 = (15 + v_0) \times 3$
  • $23.33 = 15 + v_0$
  • $v_0 = 8.33\,m/s$
• Step 2: Solve for $a$ using $v = v_0 + at$
  • $15 = 8.33 + a(6)$
  • $6.67 = 6a$
  • $a = 1.11\,m/s^2$

Questions & Discussion

• Q: How can we say that an object is moving?
  • A: We say an object is moving if its position changes relative to a fixed observer or reference point over a period of time.
• Q: Can you tell what each person sees in the figure (Relative Nature of Motion)?
  • A: Motion is relative. An observer standing on the ground (Observer A) might see a person (Observer B) inside a moving bus moving at the same speed as the bus. However, another person (Observer C) inside the bus would see Observer B as stationary. This illustrates that motion descriptors depend on the frame of reference.
• Q: What are the assumptions of the four kinematics equations?
  • A: The primary assumption is that the acceleration is constant ($a = \text{constant}$) throughout the entire duration being measured.
• Q: Practice Problem: A train attempted to stop from $43\,m/s$ over a distance of $111\,m$. What is its acceleration?
  • A: Given $v_0 = 43\,m/s$, $v = 0\,m/s$, $\Delta x = 111\,m$. Use $v^2 = v_0^2 + 2 a \Delta x$.
  • $0^2 = 43^2 + 2 a (111)$
  • $-1849 = 222 a$
  • $a = -8.33\,m/s^2$ (Deceleration).