Motion Under Gravity | Civil Engineering Optional for UPSC PDF Download

Mathematical Formulation of Motion Under Gravity

Force Due to Gravity

F = ma
For an object at rest on a horizontal surface, the force due to gravity ($FF$F) is equal to the weight ($WW$W), and the acceleration is the acceleration due to gravity ($gg$g). Therefore:
W = Mg
Here,

• W is the weight of the object (in Newtons, N).
• $M$M is the mass of the object (in kilograms, kg).
• $g$g is the acceleration due to gravity (approximately $9.8\hspace{0.17em}m/s29.8\; \backslash ,\; \backslash text\{m/s\}^2$9.8m/s2).

Equations of Motion Under Gravity

1. Velocity After Time t: 

   • v: Final velocity
   • $u$u: Initial velocity
   • $g$g: Acceleration due to gravity
   • $t$t: Time

2. Displacement After Time t:

   • $h$h: Displacement
   • u: Initial velocity
   • $g$g: Acceleration due to gravity
   • $t$t: Time

3. Velocity-Displacement Relationship: 

   • v: Final velocity
   • $u$u: Initial velocity
   • $g$g: Acceleration due to gravity
   • h: Displacement

Vertical Motion Under Gravity

Velocity of a Body Dropped from a Height
When an object is dropped from a height h, its initial velocity u is zero. The equation of motion becomes: 

Since 

Examples of Motion Under Gravity

1. Ball Thrown Upwards:

   • A ball thrown upwards will eventually fall back to the ground due to gravity.

2. Walking on the Ground:

   • Gravity keeps us anchored to the ground, enabling walking and other movements.

3. Tides:

   • Gravitational forces from the moon and sun cause the periodic rise and fall of ocean tides.

4. Daily Activities:

   • Jumping, sliding, running, and other activities are influenced by gravitational forces.

5. Celestial Motion:

   • The movement and alignment of celestial bodies, such as planets orbiting the sun, are governed by gravitational forces.

Conclusion

Understanding gravitational force and motion under gravity is essential for comprehending various physical phenomena. The mathematical formulations and equations of motion under gravity provide a basis for analyzing these phenomena in fields like physics, engineering, and everyday life activities. Gravitational force not only influences terrestrial activities but also plays a critical role in the behavior of celestial bodies in the universe.