Motion in One Dimension

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### Introduction to Motion in One Dimension Motion in one dimension, also known as rectilinear motion, describes the movement of an object along a straight line. This chapter focuses on understanding various kinematic quantities and their graphical representations. #### Key Terms - **Position (x):** Location of an object relative to an origin. - **Displacement ($\Delta x$):** Change in position ($\Delta x = x_f - x_i$). It is a vector quantity. - **Distance:** Total path length covered by an object. It is a scalar quantity. - **Speed (v):** Rate of change of distance. It is a scalar quantity. - Average Speed = Total Distance / Total Time - **Velocity ($\vec{v}$):** Rate of change of displacement. It is a vector quantity. - Average Velocity = Total Displacement / Total Time - Instantaneous Velocity = $\frac{dx}{dt}$ - **Acceleration ($\vec{a}$):** Rate of change of velocity. It is a vector quantity. - Average Acceleration = $\frac{\Delta v}{\Delta t}$ - Instantaneous Acceleration = $\frac{dv}{dt} = \frac{d^2x}{dt^2}$ - **Retardation/Deceleration:** Negative acceleration, meaning the velocity is decreasing. ### Displacement-Time (x-t) Graphs In an x-t graph, the slope represents velocity ($\frac{\Delta x}{\Delta t}$). #### 1. Body at Rest - **Description:** Position does not change with time. - **Graph:** A horizontal line parallel to the time axis. - **Slope:** Zero, indicating zero velocity. #### 2. Uniformly Moving Body (Constant Velocity) - **Description:** Displacement changes equally in equal intervals of time. - **Graph:** A straight line inclined to the time axis. - **Slope:** Constant and non-zero, indicating constant velocity. - Positive slope: Positive velocity. - Negative slope: Negative velocity. #### 3. Accelerating Body - **Description:** Velocity is changing with time. - **Graph:** A curve. - **Increasing Velocity (Positive Acceleration):** Curve bends upwards. - **Slope:** Increasing, indicating increasing velocity. - **Decreasing Velocity (Negative Acceleration/Retardation):** Curve bends downwards. - **Slope:** Decreasing, indicating decreasing velocity. #### 4. Body with Increasing Acceleration - **Description:** The rate of change of velocity is itself increasing. - **Graph:** A curve that becomes steeper and steeper more rapidly. Concave up. - **Slope:** Increases at an increasing rate. #### 5. Body with Decreasing Acceleration - **Description:** The rate of change of velocity is decreasing (but still positive). - **Graph:** A curve that becomes steeper but the rate of steepness increase is slowing down. Concave down. - **Slope:** Increases at a decreasing rate. #### 6. Body with Constant Acceleration - **Description:** Velocity changes uniformly with time. - **Graph:** A parabolic curve. - **Slope:** Continuously changing, and the slope of the tangent at any point gives the instantaneous velocity. #### 7. Body with Constant Retardation - **Description:** Velocity decreases uniformly with time. - **Graph:** A parabolic curve, opening downwards. - **Slope:** Continuously decreasing. ### Distance-Time (d-t) Graphs In a d-t graph, the slope represents speed ($\frac{\Delta d}{\Delta t}$). Distance is always non-decreasing. #### 1. Body at Rest - **Description:** Distance covered remains zero or constant if it started moving and then stopped. - **Graph:** A horizontal line parallel to the time axis (if d > 0) or on the time axis (if d = 0). - **Slope:** Zero, indicating zero speed. #### 2. Uniformly Moving Body (Constant Speed) - **Description:** Distance changes equally in equal intervals of time. - **Graph:** A straight line inclined to the time axis. - **Slope:** Constant and positive, indicating constant speed. #### 3. Accelerating Body (Increasing Speed) - **Description:** Speed is increasing. - **Graph:** A curve bending upwards. - **Slope:** Increasing, indicating increasing speed. #### 4. Retarding Body (Decreasing Speed) - **Description:** Speed is decreasing. - **Graph:** A curve bending downwards, but still non-decreasing. - **Slope:** Decreasing, indicating decreasing speed. ### Velocity-Time (v-t) Graphs In a v-t graph: - The slope represents acceleration ($\frac{\Delta v}{\Delta t}$). - The area under the curve represents displacement ($\int v dt$). #### 1. Body at Rest - **Description:** Velocity is zero. - **Graph:** A horizontal line coinciding with the time axis. - **Slope:** Zero. - **Area:** Zero displacement. #### 2. Uniformly Moving Body (Constant Velocity) - **Description:** Velocity is constant and non-zero. - **Graph:** A horizontal line parallel to the time axis (above for positive, below for negative velocity). - **Slope:** Zero, indicating zero acceleration. - **Area:** Represents displacement ($v \times t$). #### 3. Body with Constant Acceleration - **Description:** Velocity changes uniformly with time. - **Graph:** A straight line inclined to the time axis. - **Slope:** Constant and non-zero, indicating constant acceleration. - Positive slope: Positive acceleration. - Negative slope: Negative acceleration (constant retardation). - **Area:** Represents displacement. For a straight line, it's the area of a trapezoid or triangle. #### 4. Body with Increasing Acceleration - **Description:** The rate of change of velocity is increasing. - **Graph:** A curve bending upwards. - **Slope:** Increasing, indicating increasing acceleration. #### 5. Body with Decreasing Acceleration - **Description:** The rate of change of velocity is decreasing (but still positive). - **Graph:** A curve bending downwards. - **Slope:** Decreasing, indicating decreasing acceleration. #### 6. Body with Increasing Retardation - **Description:** Negative acceleration is increasing in magnitude (e.g., velocity drops faster and faster). - **Graph:** A curve bending downwards, becoming steeper. - **Slope:** Negative and decreasing, indicating increasing retardation. #### 7. Body with Decreasing Retardation - **Description:** Negative acceleration is decreasing in magnitude (e.g., velocity drops, but slower over time). - **Graph:** A curve bending upwards, becoming less steep. - **Slope:** Negative and increasing (less negative), indicating decreasing retardation. ### Speed-Time (s-t) Graphs In an s-t graph: - The slope represents the magnitude of acceleration. - The area under the curve represents total distance covered. Speed is always non-negative. #### 1. Body at Rest - **Description:** Speed is zero. - **Graph:** A horizontal line coinciding with the time axis. - **Slope:** Zero. - **Area:** Zero distance. #### 2. Uniformly Moving Body (Constant Speed) - **Description:** Speed is constant and non-zero. - **Graph:** A horizontal line parallel to the time axis. - **Slope:** Zero. - **Area:** Represents total distance ($s \times t$). #### 3. Body with Constant Acceleration (Increasing Speed) - **Description:** Speed increases uniformly with time. - **Graph:** A straight line inclined to the time axis, starting from the origin or some initial speed. - **Slope:** Constant and positive. - **Area:** Represents total distance. #### 4. Body with Constant Retardation (Decreasing Speed) - **Description:** Speed decreases uniformly with time until it becomes zero. - **Graph:** A straight line inclined downwards, ending at the time axis. - **Slope:** Constant and negative. - **Area:** Represents total distance. #### 5. Body with Increasing Acceleration (Increasing Speed) - **Description:** Speed increases at an increasing rate. - **Graph:** A curve bending upwards, becoming steeper. - **Slope:** Increasing. #### 6. Body with Decreasing Acceleration (Increasing Speed) - **Description:** Speed increases at a decreasing rate. - **Graph:** A curve bending downwards, becoming less steep. - **Slope:** Decreasing (but positive). #### 7. Body with Increasing Retardation (Decreasing Speed) - **Description:** Speed decreases at an increasing rate. - **Graph:** A curve bending downwards, becoming steeper. - **Slope:** Negative and decreasing. #### 8. Body with Decreasing Retardation (Decreasing Speed) - **Description:** Speed decreases at a decreasing rate. - **Graph:** A curve bending upwards, becoming less steep. - **Slope:** Negative and increasing (less negative). ### Acceleration-Time (a-t) Graphs In an a-t graph: - The area under the curve represents change in velocity ($\int a dt$). - The slope represents jerk ($\frac{da}{dt}$). #### 1. Body at Rest or Constant Velocity - **Description:** Acceleration is zero. - **Graph:** A horizontal line coinciding with the time axis. - **Area:** Zero change in velocity. #### 2. Body with Constant Acceleration - **Description:** Acceleration is constant and non-zero. - **Graph:** A horizontal line parallel to the time axis (above for positive, below for negative). - **Area:** Represents change in velocity ($a \times t$). #### 3. Body with Increasing Acceleration - **Description:** Acceleration is increasing. - **Graph:** A straight line inclined upwards, or a curve bending upwards. - **Slope:** Positive (constant jerk if straight line, increasing jerk if curve). #### 4. Body with Decreasing Acceleration - **Description:** Acceleration is decreasing. - **Graph:** A straight line inclined downwards, or a curve bending downwards. - **Slope:** Negative (constant jerk if straight line, decreasing jerk if curve). #### 5. Body with Increasing Retardation - **Description:** Acceleration is negative and its magnitude is increasing (e.g., -2, -4, -6 m/s²). - **Graph:** A line (or curve) in the negative acceleration region, moving further from the time axis. - **Slope:** Negative. #### 6. Body with Decreasing Retardation - **Description:** Acceleration is negative and its magnitude is decreasing (e.g., -6, -4, -2 m/s²). - **Graph:** A line (or curve) in the negative acceleration region, moving closer to the time axis. - **Slope:** Positive. ### Kinematic Equations for Uniformly Accelerated Motion These equations apply only when acceleration is constant. - **First Equation:** $v = u + at$ - Relates final velocity ($v$) to initial velocity ($u$), acceleration ($a$), and time ($t$). - **Second Equation:** $s = ut + \frac{1}{2}at^2$ - Relates displacement ($s$) to initial velocity ($u$), acceleration ($a$), and time ($t$). - **Third Equation:** $v^2 = u^2 + 2as$ - Relates final velocity ($v$) to initial velocity ($u$), acceleration ($a$), and displacement ($s$). - **Displacement in n-th second:** $s_n = u + \frac{a}{2}(2n - 1)$ - Gives the displacement covered specifically during the n-th second. #### Important Notes: - **Vector Quantities:** Displacement, velocity, and acceleration are vector quantities. Their signs indicate direction. - Positive direction (e.g., right, up). - Negative direction (e.g., left, down). - **Free Fall:** When an object falls freely under gravity, $a = g$ (approximately $9.8 \, m/s^2$ downwards). - If upward is positive, then $a = -g$. - If downward is positive, then $a = +g$. ### Graph Conversions and Interpretations #### x-t to v-t to a-t - **Slope of x-t graph** gives **instantaneous velocity** (v). - **Slope of v-t graph** gives **instantaneous acceleration** (a). #### a-t to v-t to x-t - **Area under a-t graph** gives **change in velocity** ($\Delta v$). - $v_f = v_i + \text{Area under a-t}$ - **Area under v-t graph** gives **change in displacement** ($\Delta x$). - $x_f = x_i + \text{Area under v-t}$ #### Summary Table | Graph Type | Slope Represents | Area Under Curve Represents | |------------|------------------|-----------------------------| | x-t Graph | Velocity | N/A | | v-t Graph | Acceleration | Displacement | | a-t Graph | Jerk | Change in Velocity | | d-t Graph | Speed | N/A | | s-t Graph | Magnitude of Acceleration | Total Distance | #### Common Pitfalls - **Distance vs. Displacement:** Always remember distance is scalar and displacement is vector. Area under speed-time graph is distance, area under velocity-time graph is displacement. - **Speed vs. Velocity:** Speed is magnitude of velocity. Speed can never be negative. - **Retardation:** Negative acceleration doesn't always mean retardation. If velocity is negative and acceleration is also negative, the object is speeding up in the negative direction. Retardation means speed is decreasing. - **Instantaneous vs. Average:** Slope of tangent gives instantaneous value, slope of secant gives average value. ### Relative Motion in One Dimension Relative motion describes the motion of an object as observed from another moving object. - **Relative Velocity:** - If two objects A and B are moving with velocities $v_A$ and $v_B$ respectively: - Velocity of A relative to B: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ - Velocity of B relative to A: $\vec{v}_{BA} = \vec{v}_B - \vec{v}_A = -\vec{v}_{AB}$ - **Case 1: Moving in the same direction:** $|v_{AB}| = |v_A - v_B|$ - **Case 2: Moving in opposite directions:** $|v_{AB}| = |v_A - (-v_B)| = |v_A + v_B|$ - **Relative Acceleration:** - If two objects A and B are moving with accelerations $a_A$ and $a_B$ respectively: - Acceleration of A relative to B: $\vec{a}_{AB} = \vec{a}_A - \vec{a}_B$ #### Applications - **Trains crossing:** Relative speed is used to calculate time taken. - **Boats in a river:** Velocity of boat relative to water and water relative to ground. - **Rain and Man problems:** Though often 2D, the concept of relative velocity is fundamental. ### Examples and Problem-Solving Strategies #### 1. Interpreting x-t Graphs - **Problem:** An x-t graph shows a horizontal line from t=0 to t=2s, then a straight line inclined upwards from t=2s to t=5s, then a straight line inclined downwards from t=5s to t=7s, returning to initial position. - **Interpretation:** - **0-2s:** Body at rest ($\Delta x = 0$, $v = 0$). - **2-5s:** Uniform positive velocity (constant positive slope). - **5-7s:** Uniform negative velocity (constant negative slope), returning to origin. #### 2. Interpreting v-t Graphs - **Problem:** A v-t graph shows a straight line starting from origin with positive slope until t=3s, then a horizontal line until t=6s, then a straight line with negative slope returning to v=0 at t=8s. - **Interpretation:** - **0-3s:** Constant positive acceleration (uniform increase in velocity). - **3-6s:** Constant positive velocity (zero acceleration). - **6-8s:** Constant negative acceleration (uniform decrease in velocity/retardation). - **Displacement:** Calculate area of trapezoid (0-8s) or sum of areas of triangle (0-3s), rectangle (3-6s), and triangle (6-8s). - **Distance:** Calculate total area by taking absolute values if any part of the graph is below the time axis. In this case, since all velocity is positive, distance = displacement. #### 3. Using Kinematic Equations - **Problem:** A car accelerates from rest at $2 \, m/s^2$ for $5 \, s$. What is its final velocity and displacement? - **Solution:** - Given: $u = 0$, $a = 2 \, m/s^2$, $t = 5 \, s$. - Final velocity ($v$): $v = u + at = 0 + (2)(5) = 10 \, m/s$. - Displacement ($s$): $s = ut + \frac{1}{2}at^2 = (0)(5) + \frac{1}{2}(2)(5)^2 = 0 + 25 = 25 \, m$. #### 4. Relative Motion Problem - **Problem:** Two trains A and B are moving on parallel tracks. Train A is moving at $10 \, m/s$ and train B at $15 \, m/s$. - **Case 1: Moving in the same direction.** - $v_{AB} = v_A - v_B = 10 - 15 = -5 \, m/s$. (A appears to move backwards relative to B) - $v_{BA} = v_B - v_A = 15 - 10 = 5 \, m/s$. (B appears to move forwards relative to A) - **Case 2: Moving in opposite directions.** (Let A be positive, B be negative) - $v_A = 10 \, m/s$, $v_B = -15 \, m/s$. - $v_{AB} = v_A - v_B = 10 - (-15) = 25 \, m/s$. (A appears to move away from B at 25 m/s) - $v_{BA} = v_B - v_A = -15 - 10 = -25 \, m/s$. (B appears to move away from A at 25 m/s in negative direction) ### Summary of Slopes and Areas | Graph Type | Slope Significance | Area Significance | |---|---|---| | **Displacement-Time (x-t)** | Velocity | Not directly useful for motion quantity | | **Velocity-Time (v-t)** | Acceleration | Displacement | | **Acceleration-Time (a-t)** | Jerk | Change in Velocity | | **Distance-Time (d-t)** | Speed | Not directly useful for motion quantity | | **Speed-Time (s-t)** | Magnitude of Acceleration | Total Distance | #### General Principles - A straight line in any graph indicates a constant rate of change for the quantity represented by the slope. - A curve indicates a changing rate of change. - The concavity of a curve in x-t graph indicates the sign of acceleration. - Concave up ($\cup$): Positive acceleration. - Concave down ($\cap$): Negative acceleration. - The point where the slope of an x-t graph is zero indicates a momentary rest (e.g., peak of projectile motion). - The point where the slope of a v-t graph is zero indicates constant velocity. - The point where a v-t graph crosses the time axis indicates a change in direction of motion. ### Important Distinctions and Concepts #### Speed vs. Velocity - **Speed:** Scalar quantity, always positive, total distance / total time. - **Velocity:** Vector quantity, can be positive or negative, displacement / time. - If an object moves in one direction without changing direction, speed = |velocity|. - If an object changes direction, distance > |displacement|, and average speed > |average velocity|. #### Distance vs. Displacement - **Distance:** Scalar, total path length, always non-negative. - **Displacement:** Vector, shortest path from initial to final point, can be positive, negative, or zero. - For a round trip, displacement = 0, but distance > 0. #### Acceleration vs. Retardation - **Acceleration:** Rate of change of velocity. Can be positive or negative. - **Retardation (Deceleration):** When acceleration is opposite in direction to velocity, causing the speed to decrease. - If velocity is positive and acceleration is negative, it's retardation. - If velocity is negative and acceleration is positive, it's retardation. - If velocity is positive and acceleration is positive, it's speeding up. - If velocity is negative and acceleration is negative, it's speeding up (in negative direction). #### Jerk - **Jerk:** Rate of change of acceleration. $\frac{da}{dt} = \frac{d^3x}{dt^3}$. - Represented by the slope of the a-t graph. - While less commonly used in introductory kinematics, it helps describe situations where acceleration itself is changing (e.g., in smooth stops or starts of vehicles). ### Practical Applications - **Automotive Engineering:** Designing braking systems, understanding vehicle performance. - **Sports Science:** Analyzing athlete's movements, optimizing performance. - **Traffic Management:** Predicting traffic flow, designing road infrastructure. - **Projectile Motion (simplified):** Understanding the vertical motion under gravity (though usually 2D, 1D concepts apply to each component). - **Roller Coaster Design:** Ensuring safe and thrilling rides by controlling acceleration and jerk. ### Common Mistakes to Avoid - **Confusing signs:** Always be consistent with your chosen positive and negative directions for vector quantities. - **Using kinematic equations for non-uniform acceleration:** Remember $v=u+at$ etc. only apply if 'a' is constant. - **Mixing distance and displacement:** Be clear about which quantity is required by the problem. - **Misinterpreting graph slopes/areas:** Double-check what each feature of a graph represents. - **Ignoring initial conditions:** Initial position ($x_0$), initial velocity ($u$), and initial time ($t_0$) are crucial. - **Units:** Always include units and ensure consistency (e.g., m, s, m/s, m/s²).