Uniform Acceleration and Kinetic Energy

When a body starts from rest and moves with uniform acceleration, its velocity increases at a constant rate. As the body accelerates, its kinetic energy also increases. The ratio of kinetic energy at different points in time can be determined by analyzing the relationship between velocity and kinetic energy.

Understanding Uniform Acceleration

Uniform acceleration refers to a situation where the velocity of an object changes by the same amount in equal intervals of time. In other words, the object's acceleration remains constant throughout its motion. Mathematically, uniform acceleration can be expressed as:

a = (v - u) / t

Where:
a = uniform acceleration
v = final velocity
u = initial velocity
t = time interval

Relationship between Kinetic Energy and Velocity

The kinetic energy (KE) of an object is given by the formula:

KE = 1/2 mv²

Where:
m = mass of the object
v = velocity of the object

Ratio of Kinetic Energy at Different Points in Time

To determine the ratio of kinetic energy at the end of the first, second, and third seconds of the body's motion, we need to consider the relationship between velocity and kinetic energy.

First Second:
- Let's assume the uniform acceleration of the body is 'a' during the first second.
- At the end of the first second, the velocity 'v' can be calculated using the equation:
v = u + at
- As the body starts from rest, the initial velocity 'u' is 0.
- Therefore, v = at.
- Substituting this value of velocity in the kinetic energy formula, we get:
KE₁ = 1/2 m(at)² = 1/2 mat²

Second Second:
- Now, let's consider the second second of the body's motion.
- The body continues to accelerate at a constant rate 'a'.
- At the beginning of the second second, the velocity is 'v' (which is equal to the velocity at the end of the first second).
- Using the equation v = u + at, we can find the velocity at the end of the second second:
v = u + at = at
- Substituting this value of velocity in the kinetic energy formula, we get:
KE₂ = 1/2 mat²

Third Second:
- Similarly, for the third second, the body continues to accelerate at a constant rate 'a'.
- At the beginning of the third second, the velocity is 'v' (which is equal to the velocity at the end of the second second).
- Using the equation v = u + at, we can find the velocity at the end of the third second:
v = u + at = at
- Substituting this value of velocity in the kinetic energy formula, we get:
KE₃ = 1/2 mat²

Ratio of Kinetic Energy
To find the ratio of kinetic energy at the end of the first, second, and third seconds, we can divide the kinetic energy at each interval by the previous interval.

- Ratio of KE₂ to KE₁ = KE₂ / KE₁ = (1/2 mat²) / (1/2 mat²) = 1
- Ratio of KE₃ to KE₂

V=u+at;1/2mv^2=K.E;u=0;a=constant;v'=0+a(1)=a;v"=0+a(2)=2a;V'''=0+a(3)=3a;K.E ' : K.E " : K.E ''' = v'^2 : v"^2 : v'''^2 =a^2 : (2a)^2 : (3a)^2 = 1:4:9;