Motion in a Straight Line Flashcards

Distance and Displacement

• Distance ($s$): The total path length covered by an object.
• Displacement ($\mathbf{s}$): The change in position of an object.
• General Relationship: Distance is denoted as $h + 2x$ while displacement is $-h$ in specific vertical motion scenarios.
• Curvilinear Motion: The magnitude of displacement is always less than the distance traveled in the case of curvilinear motion.
• Circular Path Calculation: If an object turns through an angle $\theta$ along a circular path of radius $r$ from point $A$ to point $B$:
  • Distance ($d$): $d = r \times \theta$
  • Displacement ($2x$): $2x = 2r \times \sin\left(\frac{\theta}{2}\right)$

Speed

• Definition: The rate at which distance is covered with respect to time is called speed.
• Nature: Scalar Quantity.
• Units:
  • S.I. Unit: Metre per second ($\text{m/s}$).
  • C.G.S. Unit: Centimetre per second ($\text{cm/s}$).
• Conversion Factor: To convert from $km/hr$ to $m/s$, multiply by $\frac{5}{18}$.
• Dimensions: $[M^0 L^1 T^{-1}]$.
• Property: For a moving particle, speed can never be zero or negative; it is always positive.

Types of Speed

• Uniform Speed: A particle covers equal distances in equal intervals of time.
• Non-uniform Speed (Variable Speed): A particle covers unequal distances in equal intervals of time.
• Average Speed: The ratio of total distance travelled to the total time taken for a given time interval.
  • Formula: $V_{avg} = \frac{\text{Total distance travelled}}{\text{Total time taken}} = \frac{\Delta s}{\Delta t}$
• Instantaneous Speed: The speed of a particle at a particular instant of time.
  • Formula: $V_{inst} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$

Mathematical Models for Average Speed

• Basic Summation: If a body travels distances $s_1, s_2, s_3$ in times $t_1, t_2, t_3$:
  • $V_{avg} = \frac{s_1 + s_2 + s_3}{t_1 + t_2 + t_3}$
• Variable Speeds over Distances: If an object travels distances $s_1, s_2, s_3$ with speeds $v_1, v_2, v_3$ respectively in the same direction:
  • $V_{avg} = \frac{s_1 + s_2 + s_3}{\frac{s_1}{v_1} + \frac{s_2}{v_2} + \frac{s_3}{v_3}}$
• Half-Journey Case: If an object travels the first half of the total journey with speed $v_1$ and the next half with speed $v_2$:
  • $V_{avg} = \frac{2 \times v_1 \times v_2}{v_1 + v_2}$
• One-Third Journey Case: If a body travels the first, second, and last $1/3^{rd}$ of the distance with speeds $v_1, v_2, v_3$ respectively:
  • $V_{avg} = \frac{3 \times v_1 \times v_2 \times v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}$
• Variable Speeds over Time Intervals: If an object travels with speeds $v_1, v_2, v_3$ during time intervals $t_1, t_2, t_3$:
  • $V_{avg} = \frac{v_1 t_1 + v_2 t_2 + v_3 t_3 + \dots}{t_1 + t_2 + t_3 + \dots}$
• Equal Time Intervals: If all time intervals are equal ($t_1 = t_2 = \dots = t_n = t$), the average speed is the arithmetic mean of individual speeds:
  • $V_{avg} = \frac{v_1 + v_2 + v_3 + \dots + v_n}{n}$

Velocity

• Definition: The rate of change of position with respect to time is called velocity.
• Nature: Vector Quantity.
• Dimensions: $[M^0 L^1 T^{-1}]$.
• Units:
  • S.I. Unit: $m/s$
  • C.G.S. Unit: $cm/s$
• Property: Velocity can be positive, negative, or zero.

Types of Velocity

• Uniform Velocity (Constant Velocity): Magnitude and direction of velocity remain same. This occurs only if the object moves in a straight line without reversing direction.
• Non-uniform Velocity (Variable): Velocity changes if either magnitude, direction, or both vary.
• Average Velocity: The ratio of net displacement to the total time taken.
  • Formula: $V_{avg} = \frac{\text{Net Displacement}}{\text{Time taken}} = \frac{\Delta \mathbf{r}}{\Delta t} = \frac{\mathbf{r}_f - \mathbf{r}_i}{t_f - t_i}$
  • Direction: It is aligned with the displacement vector.
• Instantaneous Velocity: The velocity of a particle at a particular instant of time.
  • Formula: $V_{inst} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{r}}{\Delta t} = \frac{d\mathbf{r}}{dt}$
  • Direction: Always tangential to the path followed by the particle.

Key Principles of Motion

• Speed vs. Velocity Magnitude: Average speed is always greater than or equal to the magnitude of Average Velocity (Average speed $\geq |\text{Average Bio velocity}|$).
• Uniform Velocity Condition: When a particle moves with constant velocity, the magnitude of displacement and distance covered are identical.
• Constant Speed/Variable Velocity: A particle can have constant speed but variable velocity (e.g., Uniform Circular Motion). In UCM, speed is constant at every instant, but velocity changes direction continuously.
• Instantaneous Equality: On any path, the magnitude of instantaneous velocity is equal to the instantaneous speed.
• Uniform Conditions: When moving with uniform velocity, instantaneous speed, magnitude of instantaneous velocity, average speed, and magnitude of average velocity are all equal.

Acceleration

• Definition: The rate of change of velocity is called acceleration.
• Nature: Vector quantity.
• Direction: Same as the direction of change in velocity (not necessarily in the direction of velocity itself).
• Dimensions: $[M^0 L^1 T^{-2}]$.
• Units:
  • S.I. Unit: $m/s^2$
  • C.G.S. Unit: $cm/s^2$
• Methods of Change: Velocity can be changed by changing magnitude only, direction only, or both magnitude and direction.

Types of Acceleration

• Uniform Acceleration: Resulting when both magnitude and direction of acceleration remain constant during motion.
• Non-Uniform Acceleration: Resulting when magnitude, direction, or both change.
• Average Acceleration: Ratio of total change in velocity to the total time taken.
  • Formula: $a_{avg} = \frac{\text{change in velocity}}{\text{time interval}} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\mathbf{v}_f - \mathbf{v}_i}{\Delta t}$
  • Direction: Along the change in velocity vector.
• Instantaneous Acceleration: Acceleration at a specific instant.
  • Formula (as function of time $t$): $a_{inst} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}$
  • Second Derivative: $a = \frac{d^2 \mathbf{r}}{dt^2}$ (second derivative of position vector).
  • Formula (as function of position $x$): $a_{inst} = v \times \frac{dv}{dx}$

Calculus Applications in Kinematics

• Differentiation Chain:
  • Position ($x$) $\xrightarrow{d/dt}$ Velocity ($v$) $\xrightarrow{d/dt}$ Acceleration ($a$).
• Integration Chain:
  • Acceleration ($a$) $\xrightarrow{\int dt}$ Change in Velocity ($\Delta v$) $\xrightarrow{\int dt}$ Change in Position/Displacement ($\Delta x$).
• Functions:
  • For $v-t$ relation: $a = \frac{dv}{dt}$
  • For $v-x$ relation: $a = v \times \frac{dv}{dx}$

Kinematical Equations of Motion

Applicable for motion along a straight line with uniform acceleration ($a$):

1. Velocity-Time: $v = u + at$
2. Displacement-Time: $S = x_f - x_i = ut + \frac{1}{2}at^2$
3. Position-Time: $x_f = x_i + ut + \frac{1}{2}at^2$
4. Velocity-Displacement: $v^2 - u^2 = 2as$
5. Displacement in $n^{th}$ second: $S_{n^{th}} = u + \frac{a}{2} (2n - 1)$
6. Displacement via Average Velocity: $S = \left( \frac{u + v}{2} \right) t$

Graphical Analysis: Position-Time ($x-t$) Graphs

• Significance: The slope of the $x-t$ graph represents instantaneous velocity ($v = \tan(\theta)$). Area defines no physical quantity.
• Case I: Slope $\tan(0) = 0$. Velocity is zero; the body is at rest.
• Case II: Slope is constant ($\theta$ is constant). Velocity is constant; body is in uniform motion.
• Case III: $\theta > 90^{\circ}$, $\tan(\theta)$ is negative. Velocity is negative but constant; uniform motion in opposite direction.
• Case IV (Non-Uniform):
  • If $\theta$ decreases with time: $\tan(\theta)$ decreases, velocity decreases.
  • If $\theta$ increases with time: $\tan(\theta)$ increases, velocity increases.

Graphical Analysis: Velocity-Time ($v-t$) Graphs

• Significance: The slope represents acceleration ($a = \tan(\theta)$). Area under the curve gives displacement and distance.
• Area Calculations:
  • Displacement: Area above axis minus area below ($A_1 - A_2$).
  • Distance: Total sum of areas ($A_1 + A_2$).
• Case I: $\theta = 0^{\circ}$, $\tan(0) = 0$. Acceleration is zero; velocity is constant (uniform motion).
• Case II: $\theta$ is constant. Acceleration is constant; uniformly accelerated motion.
• Case III: $\theta > 90^{\circ}$, $\tan(\theta)$ is negative and constant. Uniform retardation (negative acceleration) acting on the body.
• Case IV (Variable Acceleration):
  • If $\theta$ decreases: Acceleration decreases with time (not necessarily retardation).
  • If $\theta$ increases: Acceleration increases with time.

Graphical Analysis: Acceleration-Time ($a-t$) Graphs

• Significance: Slope defines nothing. The area under the graph gives the change in velocity ($\Delta v = v_f - v_i = \int a \, dt$).
• Case I: Horizontal line above axis indicates uniform/constant acceleration.
• Case II: Slanted line indicates uniformly increasing acceleration.

Practical Constraints and Key Points

• Non-existent Graphs:
  • Vertical lines (implying infinite speed or multiple values at the same time).
  • Graphs where time is constant while physical quantities change.
  • Negative speed or negative distance values.
• Two Velocities: In practice, a body cannot have two velocities, displacements, or accelerations simultaneously at one instant.

Advanced Applications

• Acceleration followed by Retardation: A car starts from rest, moves with constant acceleration $\alpha$ for some time, then retards uniformly at rate $\beta$ to come to rest. If total time is $T$:
  • Maximum Velocity ($v_{max}$): $v_{max} = \frac{\alpha \beta}{\alpha + \beta} \times T$
  • Total Distance ($S$): $S = \frac{1}{2} \left( \frac{\alpha \beta}{\alpha + \beta} \right) T^2$
• Calculation of Acceleration from $n^{th}$ second Displacement: If a particle starts from rest ($u=0$) and travels distances $S_n$ and $S_m$ in the $n^{th}$ and $m^{th}$ seconds:
  • $S_n = 0 + \frac{a}{2}(2n - 1)$
  • $S_m = 0 + \frac{a}{2}(2m - 1)$
  • Acceleration ($a$): $a = \frac{S_n - S_m}{n - m}$

Questions & Discussion

• Distance vs Displacement Relation: In curvilinear motion, displacement magnitude is always less than distance. They are equal only in straight-line motion without reversal.
• Uniform Circular Motion (UCM) Properties: A particle in UCM has constant speed but variable velocity due to continuous change in direction; thus, it is an accelerated motion even if speed is constant.
• Graph Area of Displacement-Time: Does the area of an $x-t$ graph represent anything? No, it has no physical significance ($\int x \, dt$).