Motion in a Straight Line — Revision Notes

⚡ 30-Second Revision

Position ($x$): — Location relative to origin (vector in 1D, sign indicates direction).
Distance: — Total path length (scalar, always $ge 0$).
Displacement ($Delta x$): — Change in position ($x_f - x_i$) (vector, can be $pm$ or $0$).
Speed: — Rate of distance covered (scalar, always $ge 0$).
Velocity ($vec{v}$): — Rate of displacement ($Delta x / Delta t$ or $dx/dt$) (vector, can be $pm$ or $0$).
Acceleration ($vec{a}$): — Rate of change of velocity ($Delta v / Delta t$ or $dv/dt$) (vector, can be $pm$ or $0$).
**Kinematic Equations (constant $a$):**

* $v = u + at$ * $s = ut + \frac{1}{2}at^2$ * $v^2 = u^2 + 2as$ * $s_n = u + \frac{a}{2}(2n - 1)$ (displacement in $n^{\text{th}}$ second)

Relative Velocity (1D): — $vec{v}_{AB} = vec{v}_A - vec{v}_B$. If same direction, subtract speeds. If opposite, add speeds (careful with signs).
Graphs:

* x-t slope = $v$; v-t slope = $a$. * v-t area = $s$; a-t area = $Delta v$.

2-Minute Revision

Motion in a straight line, or rectilinear motion, is the simplest form of kinematics. It involves understanding key terms: position (location relative to an origin), distance (total path length, scalar), and displacement (net change in position, vector). Speed is the rate of distance covered, while velocity is the rate of displacement, including direction. Acceleration describes the rate of change of velocity. Always use consistent sign conventions for direction.

For uniformly accelerated motion, the three core kinematic equations are indispensable: $v = u + at$, $s = ut + \frac{1}{2}at^2$, and $v^2 = u^2 + 2as$. Remember the special formula for displacement in the $n^{\text{th}}$ second: $s_n = u + \frac{a}{2}(2n - 1)$.

Free fall is a common application where $a=g$. Graphical analysis is crucial: the slope of a position-time graph gives velocity, and the slope of a velocity-time graph gives acceleration. The area under a velocity-time graph gives displacement, and under an acceleration-time graph gives change in velocity.

Finally, relative velocity in 1D is $vec{v}_{AB} = vec{v}_A - vec{v}_B$, simplifying problems where objects move relative to each other.

5-Minute Revision

Motion in a straight line is the foundation of kinematics. Start by solidifying the definitions: Position ($x$) is an object's location, a vector quantity in 1D indicated by its sign. Distance is the total path covered (scalar, always positive).

Displacement ($Delta x = x_f - x_i$) is the net change in position (vector, can be positive, negative, or zero). Speed is the rate of distance, while velocity ($v = dx/dt$) is the rate of displacement, incorporating direction.

Acceleration ($a = dv/dt$) is the rate of change of velocity. Crucially, always establish a positive direction and stick to it for all vector quantities.

For problems with constant acceleration, the kinematic equations are your primary tools:

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$v = u + at$ (relates final velocity, initial velocity, acceleration, and time)
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$s = ut + \frac{1}{2}at^2$ (relates displacement, initial velocity, acceleration, and time)
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$v^2 = u^2 + 2as$ (relates final velocity, initial velocity, acceleration, and displacement)
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$s_n = u + \frac{a}{2}(2n - 1)$ (displacement in the $n^{\text{th}}$ second)

Example: A car starts from rest ($u=0$) and accelerates at $2,\text{m/s}^2$ for $10,\text{s}$. Find its final velocity and displacement. *Solution:* $v = u+at = 0 + (2)(10) = 20,\text{m/s}$. $s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2}(2)(10^2) = 100,\text{m}$.

Graphical analysis is equally important. The slope of an x-t graph gives velocity; the slope of a v-t graph gives acceleration. The area under a v-t graph gives displacement; the area under an a-t graph gives change in velocity. Practice interpreting curves: a straight line on an x-t graph means constant velocity; a parabola means constant acceleration. A horizontal line on a v-t graph means constant velocity (zero acceleration).

Relative velocity is key for multi-object problems. $vec{v}_{AB} = vec{v}_A - vec{v}_B$. If two trains are moving towards each other at $v_1$ and $v_2$, their relative speed of approach is $v_1+v_2$. If they are moving in the same direction, the relative speed is $|v_1-v_2|$. Always be mindful of the frame of reference.

Prelims Revision Notes

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Basic Definitions:

* **Position ($x$):** Location relative to origin. Vector (sign indicates direction). * Distance (Path Length): Total length of path. Scalar. Always $ge 0$. * **Displacement ($Delta x = x_f - x_i$):** Change in position. Vector. Can be $pm$ or $0$. * Speed: Rate of distance. Scalar. Always $ge 0$. * **Velocity ($v = dx/dt$):** Rate of displacement. Vector. Can be $pm$ or $0$. * **Acceleration ($a = dv/dt = d^2x/dt^2$):** Rate of change of velocity. Vector. Can be $pm$ or $0$.

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**Uniformly Accelerated Motion (Constant $a$):**

* $v = u + at$ * $s = ut + \frac{1}{2}at^2$ * $v^2 = u^2 + 2as$ * $s_n = u + \frac{a}{2}(2n - 1)$ (Displacement in $n^{\text{th}}$ second) * Free Fall: $a = g$ (downwards). Use consistent sign convention (e.g., upward positive, $g = -9.8,\text{m/s}^2$).

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Graphical Analysis:

* Position-Time (x-t) Graph: * Slope = Instantaneous Velocity. * Straight line: Constant velocity. * Curve: Changing velocity (acceleration). * Horizontal line: Object at rest. * Velocity-Time (v-t) Graph: * Slope = Instantaneous Acceleration.

* Area under curve = Displacement. * Straight line (non-zero slope): Constant acceleration. * Horizontal line: Constant velocity (zero acceleration). * Line passing through origin: Constant acceleration from rest.

* Acceleration-Time (a-t) Graph: * Area under curve = Change in Velocity ($Delta v$). * Horizontal line: Constant acceleration.

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Relative Velocity (1D):

* Velocity of A with respect to B: $vec{v}_{AB} = vec{v}_A - vec{v}_B$. * If objects move in the same direction, relative speed is $|v_A - v_B|$. * If objects move in opposite directions, relative speed is $|v_A + v_B|$.

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Key Points & Traps:

* Average speed $ge$ |Average velocity|. Equality only if no change in direction. * Zero velocity does not mean zero acceleration (e.g., object at peak of projectile motion). * Negative acceleration means acceleration is in the negative direction, not necessarily slowing down (e.g., speeding up in negative direction). * Always define a positive direction and stick to it for signs of $x, v, a, s$.

Vyyuha Quick Recall

SUVAT for Kinematics: S - Displacement U - Initial Velocity V - Final Velocity A - Acceleration T - Time

Remember the equations by linking these letters: Very Useful Always To know: $V = U + AT$ Some Understand To All Things: $S = UT + \frac{1}{2}AT^2$ Very Useful Always Simple: $V^2 = U^2 + 2AS$