# Motion in One Dimension Formulas

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## Formulae Sheet for Motion in One Dimension

1. DISTANCE (s) (Scalar) = Total path cover

Length of path followed by particle = length of I path (See fig.)

2. DISPLACEMENT (r) (Vector) = final position – initial position

Shortest Length of path followed by particle = length of IInd path.
(See. fig.)

3. SPEED (scalar)

1. Instantaneous speed = $$\frac{\mathrm{ds}}{\mathrm{dt}}$$

2. Average speed $$\overline{\mathrm{V}}=\frac{\text { Total dis tan ce }}{\text { Total time }}$$
$$\overline{\mathrm{V}}=\frac{s_{1}+s_{2}+\ldots . s_{n}}{t_{1}+t_{2}+\ldots . t_{n}}=\frac{s_{1}+s_{2}+\ldots s_{n}}{\frac{s_{1}}{V_{1}}+\frac{s_{2}}{V_{2}}+\ldots . . \frac{s_{n}}{V_{n}}}=\frac{V_{1} t_{1}+V_{2} t_{2}+\ldots \ldots . V_{n} t_{n}}{t_{1}+t_{2}+\ldots \ldots \ldots \ldots . t_{n}}$$

• If t1 = t2 = …… = tn; $$\overline{\mathrm{V}}=\frac{V_{1}+V_{2}+\ldots \ldots+V_{n}}{n}$$,
for n = 2, $$\overline{\mathrm{V}}=\frac{V_{1}+V_{2}}{2}$$
• If s1 = s2 = …….. = sn; $$\bar{V}=\frac{n}{\frac{1}{V_{1}}+\frac{1}{V_{2}}+\ldots \ldots \frac{1}{V_{n}}}$$;
in case n = 2 $$\overline{\mathrm{V}}=\frac{2 \mathrm{V}_{1} \mathrm{V}_{2}}{\mathrm{V}_{1}+\mathrm{V}_{2}}$$

4. VELOCITY $$\overrightarrow{\mathrm{V}}$$ (vector)

1. Instantaneous velocity = $$\frac{\overrightarrow{\mathrm{dr}}}{\mathrm{dt}}$$

2. Average velocity = $$\frac{\text { Total displacement }}{\text { Total Time }}=\frac{\overrightarrow{\mathrm{r}}_{2}-\overrightarrow{\mathrm{r}}_{1}}{\mathrm{t}_{2}-\mathrm{t}_{1}}=\frac{\Delta \overrightarrow{\mathrm{r}}}{\Delta t}$$
= $$\left|\vec{V}_{a v}\right|$$ ≤ $$\overline{\mathrm{V}}$$

5. FOR UNIFORM MOTION

• Distance= speed × time
• Displacement = velocity × time

FOR UNIFORM ACCELERATION WHEN $$\overrightarrow{\mathbf{a}}$$ = CONST.

u → initial velocity, v → final velocity,
a → acceleration, s → displacement
Note:- Motion under gravity
v = u – $$\frac{1}{2}$$gt2
v2 = u2 – 2 gh
hnth = u – $$\frac{1}{2}$$ g(2n – 1)
(i) If a particle is thrown vertically upwards with initial velocity u then equation of motion becomes

• Maximum height attained by the particle = $$\frac{u^{2}}{2 g}$$
• Time to reach maximum height = $$\frac{\mathrm{u}}{\mathrm{g}}$$
• Time of flight = $$\frac{2 u}{g}$$

(ii) If a particle is thrown vertically downwards with initial velocity u then equation of motion becomes
v = u + gt
y =ut+ $$\frac{1}{2}$$ gt2
v2 = u2 + 2 gy
ynth = u+ $$\frac{1}{2}$$g(2n – 1)

6. ACCELERATION

1. Instantaneous acceleration
$$\overrightarrow{\mathrm{a}}=\frac{\mathrm{d} \overrightarrow{\mathrm{v}}}{\mathrm{dt}}=\frac{\mathrm{d}^{2} \overrightarrow{\mathrm{r}}}{\mathrm{dt}^{2}}, \overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{dv}}}{\mathrm{ds}}$$

2. Average acceleration
$$\vec{a}=\frac{\text { net change in velocity }}{\text { total time }}=\frac{\Delta \overrightarrow{\mathrm{v}}}{\Delta \mathrm{t}}=\frac{\overrightarrow{\mathrm{v}}_{2}-\overrightarrow{\mathrm{v}}_{1}}{\mathrm{t}_{2}-\mathrm{t}_{1}}$$

7. DISTANCE COVERED

S = $$\int_{t_{1}}^{t_{2}}$$v dt = Area under the v – t curve and t axis, (see fig.)

→ distance = Area I + Area II
→ displacement = Area I – Area II

• If acceleration (a) is the function of time then
Change in velocity = v2 – v1 = $$\int_{v_{1}}^{v_{2}}$$ dv = $$\int_{t_{1}}^{t_{2}}$$ adt = Area under the a – t curve and t axis.
• If acceleration (a) is the function of position (x) then
$$\int_{v_{1}}^{v_{2}}$$vdv = $$\int_{x_{1}}^{x_{2}}$$adx

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