10.1 Kinematics: Displacement, Velocity, Acceleration & Motion Equations

Kinematics: Displacement, Velocity, Acceleration & Motion Equations

In the Intermediate Physics Part I syllabus and the UET ECAT examination, kinematics provides the mathematical language for describing the motion of particles without considering the forces causing that motion. ECAT kinematics problems emphasize speed-arithmetic under strict no-calculator constraints, testing vector relationships between distance and displacement, derivatives of position, graphical slope-and-area interpretations, equations of uniformly accelerated motion, and vertical free-fall dynamics.

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1. Positional Vectors: Distance vs. Displacement

Motion represents a continuous change in the position of a body relative to a chosen reference frame.

Distance ($s$)

Distance is a scalar quantity representing the actual total length of the path traversed by a body during its motion between two points. Because it depends on the complete geometric route taken, distance is always non-negative ($s \ge 0$) and monotonically increases with time for a moving particle ($ds \ge 0$). Its SI unit is the meter ($\text{m}$) with fundamental dimension $[L]$.

Displacement ($\Delta\mathbf{r}$ or $\mathbf{d}$)

Displacement is a vector quantity defined as the shortest straight-line directed distance from the initial position vector $\mathbf{r}_i$ to the final position vector $\mathbf{r}_f$:

$\Delta\mathbf{r} = \mathbf{r}_f - \mathbf{r}_i = (x_f - x_i)\hat{\mathbf{i}} + (y_f - y_i)\hat{\mathbf{j}} + (z_f - z_i)\hat{\mathbf{k}}$

Displacement depends solely on the initial and final endpoints and is completely independent of the intermediate path traversed. It can be positive, negative, or zero.

The Geometric Inequality

The straight line is the shortest distance between two points in Euclidean space, yielding the fundamental relationship:

$|\Delta\mathbf{r}| \le s \implies \frac{s}{|\Delta\mathbf{r}|} \ge 1$

The magnitude of displacement equals the distance traveled ($|\Delta\mathbf{r}| = s$) if and only if the body moves along a straight line in a constant, unchanging direction without turning back.

Classical ECAT Comparative Scenarios

1. Semicircular Path: A particle travels along a semicircle of radius $R$ from point $A$ to point $B$:
   • Distance traversed: $s = \pi R$
   • Magnitude of displacement: $|\Delta\mathbf{r}| = 2R$ (the diameter)
   • Ratio: $\frac{s}{|\Delta\mathbf{r}|} = \frac{\pi R}{2R} = \frac{\pi}{2} \approx 1.57$
2. Closed Circular Orbit: A particle completes $N$ complete revolutions of radius $R$:
   • Distance traversed: $s = 2\pi R N$
   • Magnitude of displacement: $|\Delta\mathbf{r}| = 0$ (initial and final positions coincide)

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2. Speed and Velocity: Average vs. Instantaneous

Speed

Speed is the scalar rate at which distance is covered:

$\text{Average Speed} = v_{\text{avg}} = \frac{\text{Total Distance Traversed}}{\text{Total Time Elapsed}} = \frac{\Delta s}{\Delta t}$

Velocity

Velocity is the vector rate of change of displacement with respect to time:

$\text{Average Velocity} = \mathbf{v}_{\text{avg}} = \frac{\text{Net Displacement}}{\text{Total Time Elapsed}} = \frac{\Delta\mathbf{r}}{\Delta t}$

Because $|\Delta\mathbf{r}| \le s$, it follows that $|\mathbf{v}_{\text{avg}}| \le v_{\text{avg}}$.

Instantaneous Velocity

Instantaneous velocity is the limiting value of the average velocity as the time interval $\Delta t$ approaches zero, representing the instantaneous time derivative of the position vector:

$\mathbf{v} = \lim_{\Delta t \to 0} \frac{\Delta\mathbf{r}}{\Delta t} = \frac{d\mathbf{r}}{dt} = \frac{dx}{dt}\hat{\mathbf{i}} + \frac{dy}{dt}\hat{\mathbf{j}} + \frac{dz}{dt}\hat{\mathbf{k}}$

The magnitude of instantaneous velocity is identically equal to the instantaneous speed:

$v = |\mathbf{v}| = \left|\frac{d\mathbf{r}}{dt}\right| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}$

ECAT Trap Alert: While the magnitude of instantaneous velocity always equals instantaneous speed, the magnitude of average velocity is generally not equal to average speed ($|\mathbf{v}_{\text{avg}}| \ne v_{\text{avg}}$) whenever the direction of motion changes during the interval.

ECAT Speed Formulas for Average Speed

Case A: Equal Distance Intervals (Harmonic Mean)

If a body travels the first half of a total distance with speed $v_1$ and the second half with speed $v_2$:

• $t_1 = \frac{s/2}{v_1}$, $t_2 = \frac{s/2}{v_2}$
• $v_{\text{avg}} = \frac{s}{t_1 + t_2} = \frac{s}{\frac{s}{2v_1} + \frac{s}{2v_2}} = \frac{2 v_1 v_2}{v_1 + v_2}$

Case B: Equal Time Intervals (Arithmetic Mean)

If a body moves with speed $v_1$ for time interval $t$ and with speed $v_2$ for an equal time interval $t$:

• $s_1 = v_1 t$, $s_2 = v_2 t$
• $v_{\text{avg}} = \frac{s_1 + s_2}{2t} = \frac{v_1 t + v_2 t}{2t} = \frac{v_1 + v_2}{2}$

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3. Acceleration: Definitions and Calculus Formulations

Acceleration measures the time rate of change of velocity. Because velocity is a vector possessing both magnitude (speed) and direction, acceleration occurs whenever speed changes, direction changes, or both change simultaneously.

Average and Instantaneous Acceleration

$\mathbf{a}_{\text{avg}} = \frac{\Delta\mathbf{v}}{\Delta t} = \frac{\mathbf{v}_f - \mathbf{v}_i}{t_f - t_i}$

$\mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta\mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}$

In SI units, acceleration is expressed in meters per second squared ($\text{m/s}^2$ or $\text{m}\cdot\text{s}^{-2}$) with dimensions $[L T^{-2}]$.

Acceleration as a Function of Position: The Chain Rule

When acceleration is given as a function of position $x$ rather than time $t$, applying the differential chain rule yields the essential relation:

$a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}$

Integrating this differential equation allows determination of velocity from a spatially varying acceleration $a(x)$:

$\int_{v_i}^{v_f} v dv = \int_{x_i}^{x_f} a(x) dx \implies \frac{1}{2}(v_f^2 - v_i^2) = \int_{x_i}^{x_f} a(x) dx$

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4. Uniform Acceleration and Kinematic Equations

When acceleration is constant in both magnitude and direction (uniform acceleration), the equations of motion can be derived analytically.

The Three Classical Kinematic Equations

1. First Equation of Motion: $v_f = v_i + at$

2. Second Equation of Motion: $S = v_i t + \frac{1}{2}at^2$

3. Third Equation of Motion: $v_f^2 - v_i^2 = 2aS$

4. Average Velocity Form: $S = \left(\frac{v_i + v_f}{2}\right)t = v_{\text{avg}} t$

Critical Requirement: These four kinematic equations are valid strictly and exclusively when acceleration is uniform ($a = \text{constant}$). For time-dependent or position-dependent acceleration (such as simple harmonic motion where $a = -\omega^2 x$), these formulas fail completely and calculus must be used.

Distance Traveled in the $n$-th Second

A frequent ECAT question involves finding the distance traveled specifically during the $n$-th second of motion, which is the difference between displacement at $t = n$ and displacement at $t = n - 1$:

$S_n = S(n) - S(n-1) = \left[v_i n + \frac{1}{2}an^2\right] - \left[v_i(n-1) + \frac{1}{2}a(n-1)^2\right]$

Expanding and simplifying algebraically:

$S_n = v_i + \frac{1}{2}a[n^2 - (n^2 - 2n + 1)] = v_i + \frac{a}{2}(2n - 1)$

Galileo's Law of Odd Numbers

If a body starts from rest ($v_i = 0$) and accelerates uniformly:

• In 1st second ($n=1$): $S_1 = \frac{a}{2}(2(1) - 1) = 1\left(\frac{a}{2}\right)$
• In 2nd second ($n=2$): $S_2 = \frac{a}{2}(2(2) - 1) = 3\left(\frac{a}{2}\right)$
• In 3rd second ($n=3$): $S_3 = \frac{a}{2}(2(3) - 1) = 5\left(\frac{a}{2}\right)$
• In $n$-th second: $S_n = (2n - 1)\left(\frac{a}{2}\right)$

The ratio of distances traversed in consecutive equal time intervals by a body starting from rest under uniform acceleration forms the sequence of consecutive odd integers:

$S_1 : S_2 : S_3 : S_4 : \dots = 1 : 3 : 5 : 7 : \dots : (2n-1)$

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5. Graphical Analysis of Kinematics

Motion graphs provide an immediate geometric solution to kinematics problems without solving coupled algebraic equations.

1. Displacement-Time ($s-t$) Graph

• Slope of the tangent: Represents instantaneous velocity: $\text{Slope} = \frac{ds}{dt} = v$
• Horizontal straight line: Slope $= 0 \implies$ body is stationary ($v = 0$).
• Inclined straight line: Constant slope $\implies$ uniform velocity ($a = 0$).
• Parabolic curve opening upward: Slope is increasing $\implies$ positive uniform acceleration ($a > 0$).
• Parabolic curve opening downward: Slope is decreasing $\implies$ negative uniform acceleration or deceleration ($a < 0$).

2. Velocity-Time ($v-t$) Graph

• Slope of the tangent: Represents instantaneous acceleration: $\text{Slope} = \frac{dv}{dt} = a$
• Area under the curve: Represents net displacement: $\text{Area} = \int_{t_1}^{t_2} v(t) dt = \Delta s$
  • Area lying above the time axis represents positive displacement.
  • Area lying below the time axis represents negative displacement.
  • Total Distance: Calculated by summing the absolute magnitudes of the areas: $\text{Distance} = |\text{Area}_+| + |\text{Area}_-|$.
  • Net Displacement: Calculated by algebraic summation: $\text{Displacement} = \text{Area}_+ - |\text{Area}_-|$.

3. Acceleration-Time ($a-t$) Graph

• Area under the curve: Represents the net change in velocity: $\text{Area} = \int_{t_1}^{t_2} a(t) dt = v_f - v_i = \Delta v$

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6. Motion Under Gravity (Vertical Free Fall)

In the absence of air resistance, all bodies near the Earth's surface fall with the same uniform gravitational acceleration directed downward toward the center of the Earth:

$\mathbf{a} = -g\hat{\mathbf{j}} \quad (g = 9.8\text{ m/s}^2 \approx 10\text{ m/s}^2 \text{ for rapid ECAT arithmetic})$

Sign Convention

Taking the vertically upward direction as positive ($+y$) and downward as negative ($-y$):

• Acceleration is always: $a = -g$
• Velocity upward is positive ($v > 0$), velocity downward is negative ($v < 0$)
• Displacement above the release point is positive ($S > 0$), below release point is negative ($S < 0$)

Case 1: Body Dropped Freely from Rest ($v_i = 0$)

A body is released from height $h$:

• Velocity after falling through height $h$: $v_f = \sqrt{2gh}$
• Time required to strike the ground: $t = \sqrt{\frac{2h}{g}}$
• Distance fallen in time $t$: $h = \frac{1}{2}gt^2$

Case 2: Body Thrown Vertically Upward with Velocity $u$

• Velocity at maximum height: $v = 0$
• Maximum height attained ($H$): $0 - u^2 = 2(-g)H \implies H = \frac{u^2}{2g}$
• Time to reach maximum height (time of ascent $t_{\text{up}}$): $0 = u - gt_{\text{up}} \implies t_{\text{up}} = \frac{u}{g}$
• Total time of flight ($T$): $T = 2 t_{\text{up}} = \frac{2u}{g}$
• Velocity upon returning to the projection level: $v = -u$ (magnitude is identical to launch speed, directed downward).

Ascending Platform / Hot-Air Balloon Problems

When an object is released or dropped from an ascending vehicle (such as a hot-air balloon moving upward with steady speed $v_b$ at elevation $h$):

• The dropped object instantaneously retains the upward velocity of the vehicle: $v_i = +v_b$.
• Its displacement upon hitting the ground is $S = -h$.
• The governing kinematic equation is: $-h = v_b t - \frac{1}{2}gt^2 \implies \frac{1}{2}gt^2 - v_b t - h = 0$

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7. Relative Velocity in One Dimension

The relative velocity of a body $A$ with respect to a body $B$ is the vector difference between their individual velocities relative to a common reference frame (the ground):

$\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B$

• Moving in the Same Direction: Relative speed is the difference between their speeds: $v_{\text{rel}} = |v_A - v_B|$
• Moving in Opposite Directions: Relative speed is the sum of their speeds: $v_{\text{rel}} = v_A + v_B$

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Kinematic Equations and Principles Summary Table