Motion in One Dimension Study Notes

Introduction to Kinematics

Kinematics is the branch of physics that describes motion while ignoring the external agents (such as forces) that might have caused or modified that motion. In this study, we focus specifically on motion in one dimension, which occurs along a straight line. Motion is defined as a continual change in an object’s position.

Types of Motion

Motion can be classified into three primary categories:

• Translational Motion: An object moves from one point to another without rotation or vibration. An example is a car traveling along a highway.

• Rotational Motion: An object spins around an internal axis. An example is the Earth’s spin on its axis.

• Vibrational Motion: An object moves back and forth around a stable equilibrium position. An example is the movement of a pendulum.

Position, Displacement, and Distance

Position

An object’s position is its location relative to a chosen reference point.

• In one dimension, position is typically denoted by the variables $x$ or $y$.

• In the International System of Units (SI), position is measured in meters ($m$).

• In higher dimensions, position is denoted by the vector $\mathbf{r}$.

Displacement

Displacement is defined as the change in position during a specific time interval. It is represented as $\Delta x$. The formula is: $\Delta x \equiv x_f - x_i$ where:

• $x_f$ is the final position.

• $x_i$ is the initial position.

• The SI units are meters ($m$).

• $\Delta x$ can be positive or negative, indicating direction along the axis.

Distance versus Displacement

It is critical to distinguish between distance and displacement:

• Distance is the total length of the path followed by a particle. It is a scalar quantity and is always positive.

• Displacement is the net change in position. It is a vector quantity and can be zero even if the distance is large.

• Example: If a player moves from one end of a court to the other and then back to the starting point:

  • The distance is twice the length of the court.

  • The displacement is zero because $x_f = x_i$, resulting in $\Delta x = 0$.

Representation of Motion

Motion can be represented using various models:

• Pictorial: Visual drawings of the object at different times.

• Graphical: Plotting variables such as position vs. time.

• Tabular: Listing data points (time and position) in a table.

• Mathematical: Using algebraic or calculus-based equations to describe motion.

The Position-Time Graph

In a position-time graph, the motion of a particle (like a car) is plotted with time ($t$) on the horizontal axis and position ($x$) on the vertical axis. A smooth curve drawn through data points represents an estimate of the object's position between those specific data observations.

Vectors and Scalars

• Vector Quantities: These require both magnitude (numerical size) and direction for a complete description. In one-dimensional motion, directions are indicated using positive ($+$) and negative ($-$) signs.

• Scalar Quantities: these are completely described by magnitude alone and have no associated direction.

Velocity and Speed

Average Velocity

Average velocity ($v_{x, avg}$) is the rate at which displacement occurs along a specific axis (e.g., the x-axis). The dimensions are length divided by time ($[L/T]$), and the SI unit is $m/s$. Mathematically, it is the slope of the line connecting two points on a position-time graph: $v_{x, avg} \equiv \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

Average Speed

Average speed is a scalar quantity and is defined as the total distance traveled divided by the total time elapsed: $v_{avg} \equiv \frac{d}{t}$

• SI unit: $m/s$.

• Speed has no direction and is always expressed as a positive number.

• Average values do not provide details about the specific motion within the time interval.

Instantaneous Velocity and Speed

• Instantaneous Velocity: The limit of the average velocity as the time interval approaches zero. It is the derivative of position with respect to time: $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Instantaneous velocity can be positive, negative, or zero.

• Instantaneous Speed: The magnitude of the instantaneous velocity. It has no direction and is always positive.

Acceleration

Average Acceleration

Acceleration is the rate of change of velocity. The dimensions are $[L/T^2]$, and the SI unit is $m/s^2$. In one dimension, direction is indicated by positive or negative signs. $a_{x, avg} \equiv \frac{\Delta v_x}{\Delta t} = \frac{v_{xf} - v_{xi}}{t_f - t_i}$

Instantaneous Acceleration

Instantaneous acceleration is the limit of the average acceleration as $\Delta t$ approaches zero. It is the derivative of velocity with respect to time, or the second derivative of position with respect to time: $a_x = \lim_{\Delta t \to 0} \frac{\Delta v_x}{\Delta t} = \frac{dv_x}{dt} = \frac{d^2 x}{dt^2}$

Graphical Analysis of Motion

• The slope of a position-time graph yields the Velocity ($m/s$).

• The slope of a velocity-time graph yields the Acceleration ($m/s^2$).

Motion Diagrams

Visual representations of motion often use arrows to indicate vectors:

• Red arrows represent velocity.

• Purple arrows represent acceleration.

• Key Note on Direction: Negative acceleration does not automatically mean an object is slowing down. If both acceleration and velocity are negative, the object is actually speeding up in the negative direction.

Kinematic Equations for Constant Acceleration

These equations apply to a particle moving along the x-axis under constant acceleration ($a_x$):

1. Velocity as a function of time (Eq 2.13): $v_{xf} = v_{xi} + a_x t$

2. Position as a function of velocity and time (Eq 2.15): $x_f = x_i + \frac{1}{2}(v_{xi} + v_{xf})t$

3. Position as a function of time (Eq 2.16): $x_f = x_i + v_{xi} t + \frac{1}{2} a_x t^2$

4. Velocity as a function of position (Eq 2.17): $v_{xf}^2 = v_{xi}^2 + 2a_x(x_f - x_i)$

Note: If acceleration is zero ($a = 0$), then $v_{xf} = v_{xi} = v_x$, and the position equation reduces to the constant velocity model: $x_f = x_i + v_x t$

Free Fall Acceleration

An object in free fall is any object moving freely under the influence of gravity alone, regardless of its initial motion.

• Acceleration Direction: Always directed downward.

• Magnitude: The average magnitude at the Earth’s surface is $g = 9.80\,m/s^2$.

• Variable Nature: $g$ decreases with increasing altitude and varies with latitude.

• Coordinate System: Usually, upward is defined as positive. Thus, vertical acceleration $a_y = -g = -9.80\,m/s^2$.

Free Fall Scenarios

• Object Dropped: Initial velocity $v_{yi} = 0$.

• Object Thrown Downward: Initial velocity is negative (v_{yi} < 0).

• Object Thrown Upward: Initial velocity is positive (v_{yi} > 0). At the maximum height, the instantaneous velocity is zero ($v_y = 0$), but the acceleration remains $-9.80\,m/s^2$.

Kinematic Equations from Calculus

Using calculus, displacement is the area under the velocity-time curve, represented by a definite integral: $\Delta x = \int_{t_i}^{t_f} v_x(t) dt$

General integration forms:

• $v_{xf} - v_{xi} = \int_0^t a_x dt$

• $x_f - x_i = \int_0^t v_x dt$

Substituting constant acceleration into these integrals yields the standard kinematic equations:

• $v_{xf} = v_{xi} + a_x t$

• $x_f = x_i + v_{xi} t + \frac{1}{2} a_x t^2$

Worked Examples and Problems

Example: Average Speed and Velocity (Table 2.1)

Given positions of a car:

• A: $0\,s$, $30\,m$

• B: $10\,s$, $52\,m$

• C: $20\,s$, $38\,m$

• D: $30\,s$, $0\,m$

• E: $40\,s$, $-37\,m$

• F: $50\,s$, $-53\,m$

Displacement (A to F): $\Delta x = x_f - x_i = -53\,m - 30\,m = -83\,m$

Average Velocity (A to F): $v_{x, avg} = \frac{-83\,m}{50\,s} = -1.66\,m/s \approx -1.7\,m/s$

Average Speed: Total distance is $127\,m$ (summing absolute changes based on a specific path description). $v_{avg} = \frac{127\,m}{50\,s} = 2.54\,m/s \approx 2.5\,m/s$

Example 2.3: Particle under Function $x = -4t + 2t^2$

• Displacement (0 to 1s): $x(1) - x(0) = [-4(1) + 2(1)^2] - 0 = -2\,m$.

• Displacement (1 to 3s): $x(3) - x(1) = [-4(3) + 2(3)^2] - (-2) = 6 - (-2) = +8\,m$.

• Instantaneous Velocity at 2.5s: Calculated from slope or derivative $v_x = -4 + 4t$. At $t = 2.5$, $v_x = -4 + 4(2.5) = 6\,m/s$.

Example 2.10: Stone Thrown from Building ($y = 50.0\,m$, $v_i = 20.0\,m/s$)

• Time to Max Height: $v_f = v_i + at \implies 0 = 20.0 + (-9.80)t \implies t = 2.04\,s$.

• Max Height: $y = y_i + v_i t + \frac{1}{2} at^2 = 0 + (20.0)(2.04) + \frac{1}{2}(-9.80)(2.04)^2 = 20.4\,m$ (above build top).

• Velocity at Launch Level: $v_f^2 = v_i^2 + 2ay \implies v_f = -20.0\,m/s$ (moving downward).

• Velocity at t = 5.00s: $v = 20.0 + (-9.80)(5.00) = -29.0\,m/s$.

• Position at t = 5.00s: $y = 0 + (20.0)(5.00) + \frac{1}{2}(-9.80)(5.00)^2 = -22.5\,m$.

Test Rocket Problem

• Rocket fired upward with acceleration $a = 20.0\,m/s^2$ for $t = 4.00\,s$.

• Then motor off, coasts under gravity ($a = -9.8\,m/s^2$).

• Solution Result: Maximum elevation reached is 487 m.