Time-speed-distance

Time‑Speed‑Distance: A Comprehensive MIT‑Level Study Guide

(Quantitative Aptitude – “Time‑Speed‑Distance” (T‑S‑D) Problems)

1. The Classical Trivial Formula

Fundamental Relation
[ d = v,t \quad\Longleftrightarrow\quad v=\frac{d}{t}\quad\Longleftrightarrow\quad t=\frac{d}{v} ]

Assumptions

1. Uniform Motion: (v) is constant over the interval ([0,t]).
2. One‑Dimensional: Motion occurs along a straight line.
3. Inertial Frame: No external forces change the speed.

2. Derivation from First Principles

2.1 Kinematics of Uniform Motion

Let (x(t)) denote the position of a particle at time (t).
Uniform motion means

[ v = \frac{dx}{dt} = \text{constant}. ]

Integrating from (t_0) to (t),

[ x(t) - x(t_0) = v(t - t_0). ]

Setting (t_0 = 0) and (x(0)=0) (choosing the origin at the start point),

[ x(t) = v,t. ]

Since (x(t)=d) (the final position), we recover (d = v,t).

Rigorousness: This derivation uses the definition of the derivative and fundamental theorem of calculus. No hidden assumptions beyond differentiability of (x(t)).

2.2 From Distance as an Integral

For non‑uniform motion, speed (v(t)) can vary:

[ d = \int_{0}^{t} v(t'),dt'. ]

If (v(t)=v) constant, the integral reduces to (v,t).

Edge Case: If (v(t)) is discontinuous but integrable, the integral still defines (d). If (v(t)=0) over the interval, then (d=0).

3. Extensions Beyond the Basics

3.1 Relative Motion

When two observers are in relative motion with speeds (v_1,v_2) (signs denote direction), the relative speed is

[ v_{\text{rel}} = v_1 - v_2. ]

The distance covered by the first relative to the second is (d_{\text{rel}} = v_{\text{rel}},t).

Galilean Transformation
In classical mechanics, time is absolute: (t' = t).

3.2 Two‑Dimensional Motion

If the motion lies in the plane with position vector (\mathbf{r}(t) = (x(t),y(t))), the instantaneous speed is

[ v(t) = |\mathbf{\dot r}(t)| = \sqrt{\dot x^2 + \dot y^2}. ]

The scalar distance travelled (arc length) is

[ d = \int_{0}^{t}|\mathbf{\dot r}(t')|,dt'. ]

Edge Case: If the path is a circle of radius (R) traversed at constant angular speed (\omega), then (v = R\omega) and (d = R\omega t).

3.3 Constant Acceleration

With constant acceleration (a), speed evolves as

[ v(t) = v_0 + a t, ] where (v_0) is the initial speed. Integrating again,

[ d = v_0 t + \frac{1}{2} a t^2. ]

This is the familiar kinematic equation for uniformly accelerated motion.

Derivation: Direct integration from (v(t)=\dot x(t)).

4. Edge Cases & Singularities