# Accelerated Motion

• Equations of motion are \tt s = ut+\frac{1}{2}at^{2}
v = u + at
v− u2 = 2as
\tt S_{n} = u + a\left(n-\frac{1}{2}\right)
• Initial velocity of freely falling body is zero.
• Final velocity of freely falling body V = gt.
• Final velocity of freely falling body wr.to distance \tt v=\sqrt{2gh}
• Distance travelled in nth sec \tt S_{n}=0+g\left(n-\frac{1}{2}\right).
• Ratio of distances travelled in 1s , 2s , 3s .... = 1 : 4 : 9 : .......n2.
• Ratio of distances travelled in 1st 2nd 3rd = 1 : 3 : 5 .......(2n – 1).
• Body falls from height ‘h’ time of descent = \tt \sqrt{\frac{2h}{g}}
• For a vertically projected body The maximum height reached \tt H=\frac{u^{2}}{2g}
• Time of ascent  \tt Ta=\sqrt{\frac{2H}{g}}=\frac{u}{g}
• Time of descent \tt Td=\sqrt{\frac{2H}{g}}=\frac{u}{g}
• Time of flight \tt Tf=2\sqrt{\frac{2H}{g}}=\frac{2u}{g}
• If as friction in considered ta < td.
• Body always travels g/2 m during last second of upward motion.
• Equation of motion of body projected from top as tower is =\tt H= -u \ t\ +\frac{1}{2}gt^{2}
• Velocity with which it striker \tt V=\sqrt{u^{2}+2gh}
• If a body is projected vertically up with ‘u’ from tower it reaches ground with velocity ‘nu’ \tt H=\frac{u^{2}}{2g}\left(n^{2}-1\right)
• A particle projected vertically up from top of tower “t1” sec to reach the ground another particle thrown down with same velocity takes “t2” sec velocity of projection \tt u=\frac{g}{2}\left(t_{1}-t_{2}\right). Height of tower \tt h=\frac{1}{2}g\ t_{1}t_{2}.
• For oblique projection Horizontal position with time x = u cos θ t.
• For oblique projection vertical position with time \tt y=u\sin\theta\ t-\frac{1}{2}\ gt^{2}
• Equation of trajectory Y = \tt \tan\theta x-\frac{g}{2u^{2}\cos ^{2}\theta}\ x^{2}
• Time of ascent \tt Ta=\frac{u\sin\theta}{g}
• Time of flight \tt Tf=\frac{2u\sin\theta}{g}
• Maximum Height \tt H=\frac{u^{2}\sin^{2}\theta}{2g}
• Range (Horizontal) \tt R=\frac{u^{2}\sin 2\theta}{g}
• Range is maximum when θ = 45° R =\tt \frac{u^{2}}{g}
• Velocity of projectile at any instant t \tt V=\sqrt{\left(u\cos\theta\right)^2+\left(u\sin\theta-gt\right)^{2}}
• Angle made by projectile with horizontal \tt Tan \ \alpha=\left(\frac{u\sin\theta-gt}{u\cos\theta}\right)
• Vertical component of velocity at half of maximum height = \tt \sqrt{\left(u\cos\theta\right)^{2}+\left(\frac{u\sin\theta}{\sqrt{2}}\right)^2}
• If the particle passes two points situated at same height h at t1 and t2 time Then time of flight T = t1 + t2 & \tt h=\frac{1}{2}g\ t_{1}t_{2}
• Angular momentum of particle at maximum height about point of projection \tt L=mu\ \cos\theta\times\frac{u^{2}\sin^{2}\theta}{2g}
• Torque acting on a particle at max height = \tt mg\times\frac{R}{2}.
• For complementary angles range is same.
• Let h1 and h2 are heights “T1” and “T2” time of flights \tt h_{1}+h_{2}=\frac{u^{2}}{2g}\ R=4\sqrt{h_{1}h_{2}}\ R=\frac{1}{2}\ g\ T_{1}T_{2}
• Time of flight of oblique projection from inclined plane \tt T=\frac{2u\sin\left(\alpha-\beta\right)}{g\cos \beta}

• Range \tt R=\frac{u^{2}}{g\cos^{2}\beta}\left[sin\left(2\alpha-\beta\right)-\sin\beta\right]

• Time of flight of projectile thrown down the inclined plane \tt T=\frac{2u\sin\left(\alpha+\beta\right)}{g\cos \beta}
• Range down the plane \tt R=\frac{2u^{2}\cos\alpha\ sin\left(\alpha+\beta\right)}{g\cos^{2}\beta}
• Horizontal position of horizontal projectile X = ut
• Vertical position of horizontal projectile \tt y=\frac{1}{2}gt^{2}.
• Equation of projector of horizontal projectile \tt y=\frac{gx^{2}}{2u^{2}}
• Time of descent of horizontal projection \tt t=\sqrt{\frac{2h}{g}}
• Range of horizontal projection \tt R=u\sqrt{\frac{2h}{g}}
• Velocity of horizontal projectile at any instant \tt v=\sqrt{u^{2}+g^{2}t^{2}}
• Angle made by horizontal projectile after same time Tan α= \tt \frac{gt}{u}
• Velocity on reaching the ground \tt v=\sqrt{u^{2}+2gh}
• From a certain height if two bodies are projected horizontally with velocities ‘u1’ and ‘u2’ in opposite direction then time after which their velocity vectors one perpendicular  \tt t=\sqrt{\frac{u_{1}u_{2}}{g}}
• Distance between the two bodies when their velocities are perpendicular \tt d=\sqrt{\frac{u_{1}u_{2}}{g}}(u1+u2)
• Time after which their displacement vectors are perpendicular \tt 2\sqrt{\frac{u_{1}u_{2}}{g}}
• Distance between then when their displacement vectors are perpendicular \tt 2 \sqrt{\frac{u_{1}u_{2}}{g}}(u1+u2)
• If a body covers equal distances in equal time intervals the motion is said to be uniform.
• If a body covers unequal distances in equal time intervals the motion is said to be non uniform.
• Equation of trajectory of oblique projection from top of a tower. \tt H=-u\sin\theta\ T\ +\frac{1}{2}gT^{2}
• Range in the above case R = (u cos θ)T.
• At maximum height of oblique projective angle between velocity and acceleration is 90°.
• At maximum height of oblique projection from tower angle between velocity and acceleration is 90° .

### View the Topic in this video From 01:17 To 54:38

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1. Acceleration (a) = \tt \frac{Change\ in\ velocity\left(\Delta \nu\right)}{Time\ interval \left(\Delta t\right)}

2. Average acceleration : \tt \overrightarrow{a}_{a\nu}=\frac{\Delta\overrightarrow{\nu}}{\Delta t}=\frac{\overrightarrow{\nu}_{2}-\overrightarrow{\nu}_{1}}{\Delta t}

3. Instantaneous acceleration : \tt \overrightarrow{a}=\lim_{\Delta t \rightarrow 0}\frac{\Delta \overrightarrow{\nu}}{\Delta t}=\frac{d\overrightarrow{\nu}}{dt}

4. Equations of Uniformly Accelerated Motion:
(i) v = u + at (ii) \tt s=ut+\frac{1}{2}at^{2} (iii) v2 = u2 + 2as

5. Distance travelled in nth second \tt S_{n}=u+\frac{a}{2}\left(2n-1\right)

6. If a body moves with uniform acceleration and velocity changes from u to v in a time interval, then the velocity at the mid point of its path = \tt \frac{\sqrt{u^{2}+\nu^{2}}}{2}