Motion Along a Straight Line

Review of Vectors and Scalars

• Magnitude: Refers to the size or "how much" of a measurement.

• Vectors: Measurements that include both magnitude and direction.

• Scalars: Measurements that include only magnitude and have no direction.

• Measurement Examples:

  • Temperature: "It's $60^{\circ}F$ outside." This is a Scalar quantity.

  • Force: "I pushed with $100\,N$ north." This is a Vector quantity.

  • Distance: "I walked for $10\,m$." This is a Scalar quantity.

  • Displacement: "I walked $10\,m$ east." This is a Vector quantity.

  • Speed: "I drove at $80\,mph$." This is a Scalar quantity.

  • Velocity: "I drove $80\,mph$ west." This is a Vector quantity.

Introduction to Velocity and Speed

• Distance ($d$): A scalar quantity describing how far an object has moved along its actual path.

• Displacement ($\Delta x$): A vector quantity describing the change in position from the starting point to the ending point.

• Speed: A scalar quantity describing how fast something moves. It is always positive or zero (cannot be negative).

  • Formula: $\text{Speed} = \frac{\text{total distance}}{\text{time elapsed}}$

  • Units: $[m/s]$

• Velocity: A vector quantity describing speed in a specific direction. It can be positive or negative depending on the direction of travel relative to the origin.

  • Formula: $\text{Velocity} = \frac{\Delta x}{\Delta t} = \frac{\text{Displacement}}{\text{time elapsed}}$

  • Units: $[m/s]$

• Negative Velocity: Indicates motion in the opposite direction to the established positive direction.

• Total Trip Example: You jog $15\,m$ in $2\,s$, then jog $9\,m$ backwards in another $2\,s$.

  • Total Distance: $15\,m + 9\,m = 24\,m$.

  • Total Time: $2\,s + 2\,s = 4\,s$.

  • Speed: $\frac{24\,m}{4\,s} = 6\,m/s$.

  • Net Displacement: $15\,m - 9\,m = 6\,m$.

  • Velocity: $\frac{6\,m}{4\,s} = 1.5\,m/s$.

Solving Constant and Average Velocity Problems

• Average Velocity ($v_{avg}$): Velocity measured between two points ($initial \rightarrow final$).

• Behavior: Under conditions of constant motion with no acceleration ($a=0$), average velocity behaves exactly like constant velocity.

• Primary Equation: $v_{avg} = v = \frac{\Delta x}{\Delta t}$

• Derived Equations:

  • $\Delta x = v_{avg} \times \Delta t$

  • $x = x_0 + v_{avg} \times \Delta t$

• Example Variables:

  • If $x_0 = 0\,m$, $x = 20\,m$, and $\Delta t = 5\,s$, then $v = \frac{20\,m - 0\,m}{5\,s} = 4\,m/s$.

  • If $x_0 = 2\,m$, $\Delta t = 6\,s$, and $v = 3\,m/s$, then $x = 2 + (3 \times 6) = 20\,m$.

  • If $x = -20\,m$, $t_f = 7\,s$, $v = -4\,m/s$, and $t_0 = 3\,s$, then $\Delta t = 4\,s$ and $\Delta x = -4 \times 4 = -16\,m$. Therefore, $x_0 = -20 - (-16) = -4\,m$.

Constant Velocity with Multiple Parts

• For problems involving multiple intervals of motion, each part is calculated using the single constant velocity equation $v = \frac{\Delta x}{\Delta t}$.

• Problem-Solving Steps:

  1. Draw a diagram and list variables for each interval.

  2. Write equations for each interval.

  3. Solve for unknown variables.

• Scenario Example: A car travels at $50\,m/s$ forward for $10\,s$, then at $30\,m/s$ for $600\,m$.

  • Part 1: $\Delta x_1 = 50\,m/s \times 10\,s = 500\,m$.

  • Part 2: $\Delta t_2 = \frac{600\,m}{30\,m/s} = 20\,s$.

  • Total Distance: $500\,m + 600\,m = 1100\,m$.

  • Total Time: $10\,s + 20\,s = 30\,s$.

  • Average Velocity: $v_{avg} = \frac{1100\,m}{30\,s} \approx 36.67\,m/s$.

Introduction to Acceleration

• Definition: Acceleration is the rate at which velocity changes over time.

• Equation: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

• Units: $[m/s^2]$

• Nature of Acceleration: Acceleration is always a vector. There is no scalar equivalent (unlike distance/displacement or speed/velocity).

• Causes of Acceleration:

  1. A change in the magnitude (speed) of the velocity.

  2. A change in the direction of the velocity.

• Calculation Example: Jogging right (positive) at $6\,m/s$, and $3\,s$ later jogging left (negative) at $6\,m/s$ .

  • $\Delta v = v_f - v_0 = -6\,m/s - 6\,m/s = -12\,m/s$.

  • $a = \frac{-12\,m/s}{3\,s} = -4\,m/s^2$.

  • Magnitude is $4\,m/s^2$ and direction is to the left.

Position-Time Graphs and Velocity

• X-axis: Time ($t$).

• Y-axis: Position ($x$).

• Slope: The slope of a position-time graph represents the object's velocity ($v = \frac{rise}{run} = \frac{\Delta x}{\Delta t}$).

  • Upward Slope: Object is moving forward (positive velocity).

  • Horizontal/Flat Slope: Object is stopped (zero velocity).

  • Downward Slope: Object is moving backward (negative velocity).

  • Steeper Slopes: Represent higher velocity magnitudes.

  • Flatter Slopes: Represent lower velocity magnitudes.

• Average Velocity ($v_{avg}$): The slope of the line connecting any two specific points on the graph.

• Instantaneous Velocity ($v$): The velocity at one single point in time, determined by the slope of the tangent line at that point.

  • The velocity is zero at the peaks and valleys of the graph (where the curve turns around).

Acceleration in Motion Graphs

• Curved Position-Time Graphs: A curve indicates that velocity is changing, which means acceleration is not zero.

  • Curving UP (Smiley Face shape ☺): Indicates Positive Acceleration ($+a$).

  • Curving DOWN (Frowny Face shape ☹): Indicates Negative Acceleration ($-a$).

• Interpreting Curvature:

  • On the left side of a curve, the object may be slowing down; on the right side, it may be speeding up (or vice versa), depending on the slope of the curve.

  • Straight Line: Indicates constant velocity ($v = constant$) and zero acceleration ($a = 0$).

Velocity-Time Graphs

• X-axis: Time ($t$).

• Y-axis: Velocity ($v$).

• Slope: The slope of a velocity-time graph represents the object's acceleration ($a = \frac{\Delta v}{\Delta t}$).

  • Steeper slopes indicate higher acceleration magnitudes.

• Area Under the Curve: The area between the graph line and the time axis represents the displacement ($\Delta x$).

  • Area Above Time Axis: Positive displacement.

  • Area Below Time Axis: Negative displacement.

  • Rectangles: $Area = base \times height$.

  • Triangles: $Area = \frac{1}{2} \times base \times height$.

Acceleration-Time Graphs

• Area Under the Curve: The area between the acceleration graph line and the time axis represents the change in velocity ($\Delta v$).

  • Area Above Time Axis: Result in a positive $\Delta v$.

  • Area Below Time Axis: Result in a negative $\Delta v$.

• Final Velocity: To find the final velocity ($v_f$), you must add the initial velocity ($v_0$) to the $\Delta v$ calculated from the area: $v_f = v_0 + Area$.

Calculus in Kinematics

• Position Function ($x(t)$): An equation that provides position for any given value of time ($t$).

• Derivative Relationships:

  • Velocity is the derivative of position: $v(t) = \frac{dx}{dt}$.

  • Acceleration is the derivative of velocity: $a(t) = \frac{dv}{dt}$.

  • Power Rule: $\frac{d}{dt}(Ct^N) = N \times C t^{N-1}$.

  • Constant Rule: $\frac{d}{dt}(\text{constant}) = 0$.

• Integral Relationships:

  • Displacement is the definite integral of velocity: $\Delta x = \int_{t_0}^{t_f} v(t) \,dt$.

  • Position is the indefinite integral of velocity with an integration constant ($+C$): $x(t) = \int v(t) \,dt + C$.

  • To solve for $C$, plug in a known position and time (initial conditions).

  • Change in velocity is the definite integral of acceleration: $\Delta v = \int_{t_0}^{t_f} a(t) \,dt$.

  • Velocity is the indefinite integral of acceleration: $v(t) = \int a(t) \,dt + C$.

Equations of Motion (Kinematics Equations)

• Condition: These equations, known as Uniformly Accelerated Motion (UAM) equations, can only be used when acceleration ($a$) is constant.

• The Five Variables:

  1. $\Delta x$ (Displacement)

  2. $v_0$ (Initial velocity)

  3. $v$ (Final velocity)

  4. $a$ (Acceleration)

  5. $t$ (Time)

• The Four UAM Equations:

  1. $v = v_0 + at$ (Missing variables: $\Delta x$)

  2. $v^2 = v_0^2 + 2a\Delta x$ (Missing variables: $t$)

  3. $\Delta x = v_0t + \frac{1}{2}at^2$ (Missing variables: $v$)

  4. $\Delta x = \left(\frac{v_0 + v}{2}\right)t$ (Missing variables: $a$)

• Strategy: Identify three known variables, locate the target variable, and pick the equation that does not include the "ignored" variable (the one neither given nor asked for).

Sign Conventions and the Effect of Acceleration

• Positive Acceleration: Velocity is becoming more positive (moving toward the right or upwards).

• Negative Acceleration: Velocity is becoming more negative (moving toward the left or downwards).

• Speeding Up vs. Slowing Down:

  • Speeding Up: Occurs when velocity and acceleration have the same sign (both positive or both negative). The magnitude of velocity increases.

  • Slowing Down: Occurs when velocity and acceleration have opposite signs (e.g., $+v$ and $-a$, or $-v$ and $+a$). The magnitude of velocity decreases.

Vertical Motion and Free Fall

• Definition: An object is in free fall if the only force acting on it is gravity ($F_G$).

• Acceleration ($a_y$): All objects in free fall on Earth accelerate downwards at a rate of $g = 9.8\,m/s^2$, regardless of their mass or weight.

• Sign Convention: Usually, we define the upward direction as positive ($+$). Therefore, $a_y = -g = -9.8\,m/s^2$.

• Vertical Equations: The UAM equations are adjusted for the y-axis:

  1. $v_y = v_{0y} + a_y t$

  2. $v_y^2 = v_{0y}^2 + 2a_y \Delta y$

  3. $\Delta y = v_{0y}t + \frac{1}{2}a_y t^2$

  4. $\Delta y = \left(\frac{v_{0y} + v_y}{2}\right)t$

Catch-Up or Overtake Problems

• Condition: One object catches up to another when they are at the same position ($x_A = x_B$) at the same time ($t$).

• Solution Steps:

  1. Draw a diagram and list known variables for both objects.

  2. Write the full position equations for each object ($x = x_0 + v_0t + \frac{1}{2}at^2$).

  3. Set the equations equal to each other ($x_A = x_B$).

  4. Solve for time ($t$) and then any additional required variables like position or final velocity.

• Example: If Car A is at $x=0$ at $50\,m/s$ constant velocity and Car B is $280\,m$ ahead at $36\,m/s$ constant velocity:

  • $x_A = 0 + 50t$

  • $x_B = 280 + 36t$

  • $50t = 280 + 36t \rightarrow 14t = 280 \rightarrow t = 20\,s$.