Question 1

?How does calculus help in understanding the motion of objects under constant acceleration

Accepted Answer

Calculus is essential in understanding the motion of objects under constant acceleration because it allows us to derive and analyze the kinematic equations that describe such motion. Using calculus, we can integrate the acceleration function to obtain the velocity function, and then integrate the velocity function to find the position function. For example, if an object is accelerating at a constant rate 
𝑎
a, the velocity 
𝑣
(
𝑡
)
v(t) can be found by integrating the acceleration:
𝑣
(
𝑡
)
=
∫
𝑎
 
𝑑
𝑡
=
𝑎
𝑡
+
𝑣
0
v(t)=∫adt=at+v 
0
​
 
where 
𝑣
0
v 
0
​
  is the initial velocity. Further, integrating the velocity function gives the position 
𝑥
(
𝑡
)
x(t):
𝑥
(
𝑡
)
=
∫
𝑣
(
𝑡
)
 
𝑑
𝑡
=
∫
(
𝑎
𝑡
+
𝑣
0
)
 
𝑑
𝑡
=
1
2
𝑎
𝑡
2
+
𝑣
0
𝑡
+
𝑥
0
x(t)=∫v(t)dt=∫(at+v 
0
​
 )dt= 
2
1
​
 at 
2
 +v 
0
​
 t+x 
0
​
 
where 
𝑥
0
x 
0
​
  is the initial position. This process illustrates how calculus provides a systematic way to derive these relationships and analyze motion in a precise manner.

Question 2

.Explain the concept of electric flux and how Gauss's law uses calculus to relate it to electric fields

Accepted Answer

Electric flux is a measure of the number of electric field lines passing through a given surface. Mathematically, it is defined as the surface integral of the electric field 
𝐸
E over a surface 
𝑆
S:
Φ
𝐸
=
∫
𝑆
𝐸
⋅
𝑑
𝐴
Φ 
E
​
 =∫ 
S
​
 E⋅dA
where 
𝑑
𝐴
dA is a vector representing an infinitesimal element of the surface area with a direction normal to the surface.

Gauss's law relates electric flux to the charge enclosed by a surface. It states that the total electric flux through a closed surface is equal to the enclosed charge 
𝑄
Q divided by the permittivity of free space 
𝜖
0
ϵ 
0
​
 :
Φ
𝐸
=
∮
𝑆
𝐸
⋅
𝑑
𝐴
=
𝑄
enc
𝜖
0
Φ 
E
​
 =∮ 
S
​
 E⋅dA= 
ϵ 
0
​
 
Q 
enc
​
 
​

Using calculus, Gauss's law allows us to calculate the electric field for symmetrical charge distributions by evaluating the surface integral. For example, for a point charge 
𝑞
q inside a spherical surface of radius 
𝑟
r, the electric field is radially symmetric, and Gauss's law simplifies the calculation by using the symmetry to evaluate the integral, leading to:
𝐸
⋅
4
𝜋
𝑟
2
=
𝑞
𝜖
0
E⋅4πr 
2
 = 
ϵ 
0
​
 
q
​
 
𝐸
=
𝑞
4
𝜋
𝜖
0
𝑟
2
E= 
4πϵ 
0
​
 r 
2
 
q
​

This shows how calculus is used to connect electric fields and charges in a fundamental way.

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