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 = \(\frac{\text { Total displacement }}{\text { Total Time }}=\frac{\overrightarrow{\mathrm{r}}_{2}-\overrightarrow{\mathrm{r}}_{1}}{\mathrm{t}_{2}-\mathrm{t}_{1}}=\frac{\Delta \overrightarrow{\mathrm{r}}}{\Delta t}\)
= \(\left|\vec{V}_{a v}\right|\) Γëñ \(\overline{\mathrm{V}}\)
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
ΓåÆ 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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