# Motion Under Gravity

Whenever a particle is thrown up or down or released from a height, it falls freely under the effect of gravitational force of earth.

# Equations of motion

*v*=*u*+*gt**v*^{2}=*u*^{2}+ 2*g*(*h*â€“*h*_{0}) or*v*^{2}=*u*^{2}+ 2*gs*

where

*h*= vertical displacement,*h*_{n}_{th}= vertical displacement in*n*th second.Following are the important cases of interest:

# Case I

A particle is projected from ground with velocity

*u*in vertically upward direction, then- Time of ascent = Time of descent
- Maximum height attained =
*u*^{2}/2*g* - Speed of particle when it hits the ground =
*u* - Graphs

(a)

(b)

(c)

(d)

(e)

**Fig. 3**

- Displacement of particle in complete journey = zero â‡’ average velocity,
*v*_{av} - Distance covered by particle in complete journey

# Case II

A body is thrown upward such that it takes

*t*seconds to reach its highest point.- Distance traveled in (
*t*)th second = distance traveled in (*t*+ 1)th second. - Distance traveled in (
*t*â€“ 1)th second = distance traveled in (*t*+ 2)th second. - Distance traveled in (
*t*â€“*r*)th second = distance traveled in (*t*+*r*+ 1)th second.

# Case III

A body is projected upward from certain height

*h*with initial speed*u*.- Its speed when it acquires the height at same level is
*u*. - Its speed at the ground level is
- The time required to attain the same level is
*T*= 2*u*/*g*.

- Total time of flight (
*T*) is obtained by solving*â€“h*= +*uT*+ ,

# Some Important Notes

- From the top of a tower a body is projected upward with a certain speed; the second body is thrown downward with same speed and the third is let to fall freely from same point. Then
*t*_{1}= time of flight of body projected upward*t*_{2}= time of flight of body thrown downward*t*_{3}= time of flight of body dropped. - In case of air resistance, the time of ascent is less than time of descent for a body projected vertically upward.
- For a body projected vertically upwards, the magnitude of velocity at any given point on the path is same whether the body is moving in upward or downward direction.
- A body returns to its point of projection with the same magnitude of the velocity with which it was thrown vertically upward, provided air resistance is neglected.
- All bodies fall freely with the same acceleration.
- The acceleration of the falling bodies does not depend on the mass of the body.
- If two bodies are dropped from the same height, they reach the ground in the same time and with the same velocity.
- If a body is thrown upward with velocity
*u*from the top of a tower and another body is thrown downward from the same point and with the same velocity, then both reach the ground with the same speed. - If the friction of air be taken into account, then the motion of the object thrown upward will have the following properties:
- Time taken to go up (ascent) < time taken to come down (descent)
- The speed of the object on return to the ground is less than the initial speed. Same is true for velocity (magnitude), momentum (magnitude), and kinetic energy.
- Maximum height attained is less than
*u*^{2}/2*g*.

- A part of the kinetic energy is used up in overcoming the friction.
- A particle is dropped vertically from rest from a height. The time taken by it to fall through successive distance of 1 m each will then be in the ratio of the difference in the square roots of the integers, i.e.,