# 4.11 Projection in gravitational field

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Acceleration due to gravity is not a constant away from Earth’s surface

Gravitational force of attraction is a binding force. An object requires certain minimum velocity to break free from this attraction. We are required to impart object with certain kinetic energy to enable it to overcome gravitational pull. As the object moves away, gravitational pull becomes smaller. However, at the same time, speed of the object gets reduced as kinetic energy of the object is continuously transferred into potential energy. Remember, potential energy is maximum at the infinity.

Depending on the initial kinetic energy imparted to the projectile, it will either return to the surface or will move out of the Earth’s gravitational field.

The motion of a projectile, away from Earth’s surface, is subjected to variable force – not a constant gravity as is the case for motion near Earth’s surface. Equivalently, acceleration due to gravity, “g”, is no more constant at distances thousands of kilometers away. As such, equations of motion that we developed and used (like v = u+at) for constant acceleration is not valid for motion away from Earth.

We have already seen that analysis using energy concept is suitable for such situation, when acceleration is not constant. We shall, therefore, develop analysis technique based on conservation of energy.

## Context of motion

We need to deal with two forces for projectile : air resistance i.e. friction and gravitational force. Air resistance is an external non-conservative force, whereas gravity is an internal conservative force to the "Earth-projectile" system. The energy equation for this set up is :

${W}_{F}=\Delta K+\Delta U$

Our treatment in the module, however, will neglect air resistance for mathematical derivation. This is a base consideration for understanding motion of an object in a gravitational field at greater distances. Actual motion will not be same as air resistance at higher velocity generates tremendous heat and the projectile, as a matter of fact, will either burn up or will not reach the distances as predicted by the analysis. Hence, we should keep this limitation of our analysis in mind.

Nevertheless, the situation without friction is an ideal situation to apply law of conservation of energy. There is only conservative force in operation on the object in translation. The immediate consequence is that work by this force is independent of path. As there is no external force on the system, the changes takes place between potential and kinetic energy in such a manner that overall change in mechanical energy always remains zero. In other words, only transfer of energy between kinetic and gravitational potential energy takes place. As such,

$⇒\Delta K+\Delta U=0$

## Change in potential energy

Earlier, we used the expression “mgh” to compute potential energy or change in potential energy. We need to correct this formula for determining change in potential energy by referring calculation of potential energy to infinity. Using formula of potential energy with infinity as reference, we determine the potential difference between Earth’s surface and a point above it, as :

#### Questions & Answers

List the application of projectile
Luther Reply
How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
Abdhool Reply
show that 1w= 10^7ergs^-1
Lawrence Reply
what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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