Maths3 project

BY: -NISHANT KATARIYA
-RONAK GHODASARA

TOPICS
(1)Radio carbon dating
(2)Escape velocity
(3)Heating problems

1.RADIO CARBON DATING

 Carbon dating can be used on material which was living in the
last few tens of thousands of years, and which took its carbon
from the air. The method has become more accurate in the last
few decades.
 Carbon 14 (C14) is a radioactive isotope of carbon.
 It is produced in the upper atmosphere by radiation from the sun.
 (Specifically, neutrons hit nitrogen-14 atoms and transmute them
to carbon.)
 Land plants, such as trees, get their carbon from carbon dioxide
in the air. So, some fraction of their carbon is C14. The same is
true of any creature that gets its carbon by eating such plants.
We can measure this in living things today.

Calculation:

The number of decays per time is proportional to the current number of
radioactive atoms. This is expressed by the following differential equation,
where N is the number of radioactive atoms and λ is a positive number called
the decay constant:

This is first order ordinary differential equation in t.
Here N is dependant variable and t is a independent variable.
ƛ is a real constant.
Now we can solve it by separable variable form method.

∴ ⇒ Writing the equation in separable variable form

∴ ⇒ Integrating both the sides of equation

∴ ⇒ Simplifying the equation

∴ ⇒ Now apply initial condition at

∴ t=0
N=N0
∴

∴

∴ ⇒ Simplifying the equation after substitution of k

∴ ⇒ This is the final solution of the equation.

⇒ The mathematical model of the process of radioactivity decay is

⇒Here No=the initial amount of C14
⇒By definition, the half life is the time after which the amount of
radioactive substance has decreased to half its original value.

⇒The final solution is

1.ESCAPE VELOCITY PROBLEM

 escape velocity is the speed at which the kinetic energy plus the
gravitational potential energy of an object is zero. It is the speed needed
to "break free" from a gravitational field without further propulsion.

 Projectiles A and B fall back to earth. Projectile C achieves a circular orbit,
D an elliptical one. Projectile E escapes.

DIFFERENTIAL EQUATION
⇒ By Newton’s law of gravitation the gravitation force is proportional
to 1/r^2.

⇒ Where r is the distance from the center of the earth to the projectile.
⇒ The corresponding acceleration is
-------------------(a)

⇒ Where R=radius of the earth
g=gravitational acceleration at the earth’s surface

⇒ NOTE:
The minus sign occurs because the gravitational attraction acts in the
negative r direction i.e. towards the center of the earth

⇒ From (a) we get differential equation for the velocity v by chain rule

⇒ Substituting the last expression into (a)

⇒ This equation Ordinary differential equation in r. Here v is dependent
variable and r is independent variable and other is constants.

SOLUTION
⇒We are solving differential equation by separation of variable method

by integrating both side

⇒From the initial conditions
on the surface of the earth r=R and v=v0

⇒From the initial conditions
on the surface of the earth r=R and v=v0

⇒ From Equation (b)

DETERMINATION OF THE VELOCITY OF ESCAPE
⇒If v^2=0 then v=0 and r⟶∝ thus the projectile will stop and return to
the earth.

then vo^2 -2gR=0 and V^2 remains positive
⇒The quantity vo is called the velocity of escape from the earth.

Numerical value
radius of the earth R=6372km
gravitational acceleration g=0.0098km/sec^2

Maths3 project

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