Coupon Accepted Successfully!


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
  • 24856.png or 24865.png
  • v2 = u2 + 2g(h – h0) or v2 = u2 + 2gs
  • 24871.png
where h = vertical displacement, hnth = vertical displacement in nth 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 = u2/2g
  • Speed of particle when it hits the ground = u
  • Graphs
Fig. 3
  • Displacement of particle in complete journey = zero ⇒ average velocity, vav = 0.
  • Distance covered by particle in complete journey
    ⇒ Average speed in complete journey = 24911.png

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 – 1)th second = distance traveled in (t + 2)th second.
  • Distance traveled in (t – 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 24920.png
  • The time required to attain the same level is T = 2u/g.
  • Total time of flight (T) is obtained by solving
    –h = +uT 24941.png + 24950.png,
    where 24956.png.

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
    where, t1 = time of flight of body projected upward
    t2 = time of flight of body thrown downward
    t3 = 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 u2/2g.
  • 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.,

Test Your Skills Now!
Take a Quiz now
Reviewer Name