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Updated on 19th June, 2023 , 4 min read
The relation between kinetic energy and momentum lies in their dependence on velocity. Momentum is defined as the product of an object's mass and velocity, indicating that any object with mass "m" in motion possesses momentum. The amount of momentum an object has is determined by its mass and velocity. Hence, all objects have potential energy, which gets converted into kinetic energy when they start moving. For instance, a ball placed at the edge of a slope has potential energy, which gets converted into kinetic energy when it starts rolling down. Therefore, the mass of a moving object is referred to as "momentum," while the energy possessed by an object in motion is called "kinetic energy."
Kinetic energy refers to the energy acquired by an object due to its motion. It represents the amount of work needed to accelerate a specific mass from rest to a particular velocity. Once an object gains kinetic energy through acceleration, it retains that energy as long as its speed remains constant. When the object decelerates and comes to a stop, it will expend the same amount of work that was required to accelerate it. Additionally, any term in the Lagrangian of a system that has a time derivative is considered kinetic energy.
In mechanics, the kinetic energy of an object with mass 'm' that is moving at a speed 'v' is given by:
K.E. = ½ m. v2
Where,
m is the mass of the object, measured in kg.
v is the velocity of the object, measured in m/s.
This formula is a valid approximation in mechanics only when the speed 'v' is less than the speed of light.
In Newtonian physics, the vector quantity obtained by multiplying an object's mass with its velocity is known as linear momentum or translational momentum, simply referred to as momentum. It has both direction and magnitude and is measured in kilogram-meters per second (kgm/s) or newton-seconds in the International System of Units (SI).
The momentum of a moving object can be mathematically expressed as –
p = m.v
Where,
The momentum of an object is directly proportional to both its mass and velocity, meaning that an increase in either parameter will lead to a corresponding increase in momentum. Similarly, the kinetic energy of an object, which is the energy it possesses due to its motion, is also directly proportional to the mass and the square of the velocity of the object.
We know that,
K.E. = ½ m . v2
And, p = m. v
So we can write Kinetic Energy as:
K.E. = ½ (m . v) . v
Therefore,
K.E. = ½ p . v
Also, Kinetic Energy can be written as:
K.E. = ½ m. v2. (m / m)
è K.E. = ½ (m2. v2) / m
Therefore,
K.E. = p2 / 2m
Or we can also write:
p = √{2m(K.E.)}
To better understand the relation between kinetic energy and momentum, let's look at some examples of their calculations.
|
Object |
Mass (kg) |
Velocity (m/s) |
Kinetic Energy (J) |
Momentum (kg*m/s) |
|
Car |
1000 |
20 |
200,000 |
20,000 |
|
Baseball |
0.145 |
30 |
65.25 |
4.35 |
|
Electron |
9.11e-31 |
2.2e6 |
2.00e-14 |
1.99e-24 |
From the table, we can see that for the same object, increasing its velocity will increase both its kinetic energy and momentum. Additionally, objects with larger masses have large momenta but not necessarily larger kinetic energies.
Kinetic energy and momentum are important concepts in many areas of physics and engineering. For example:
Solution:
Momentum = mass x velocity
P = 0.5 kg x 10 m/s = 5 kg m/s
Kinetic Energy = 1/2 x mass x velocity^2
KE = 1/2 x 0.5 kg x (10 m/s)^2 = 25 J
Solution:
Velocity = momentum / mass
v = 2000 kg m/s / 500 kg = 4 m/s
Kinetic Energy = 1/2 x mass x velocity^2
KE = 1/2 x 500 kg x (4 m/s)^2 = 4000 J
Solution:
Momentum = mass x velocity
P = 10 g x 400 m/s = 4 kg m/s
(Note: 1 g = 0.001 kg)
Kinetic Energy = 1/2 x mass x velocity^2
KE = 1/2 x 0.01 kg x (400 m/s)^2 = 800 J
Solution:
Momentum = mass x velocity
P = 2 kg x 6 m/s = 12 kg m/s
Kinetic Energy = 1/2 x mass x velocity^2
KE = 1/2 x 2 kg x (6 m/s)^2 = 36 J
Solution:
Momentum = mass x velocity
P = 10000 kg x 20 m/s = 200000 kg m/s
Kinetic Energy = 1/2 x mass x velocity^2
KE = 1/2 x 10000 kg x (20 m/s)^2 = 2000000 J
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By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readThe kinetic energy and momentum of an object are both related to its velocity. Kinetic energy is the energy an object has due to its motion, while momentum is the product of an object’s mass and velocity.
The formula for kinetic energy is KE = 1/2 * mv^2, where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
The formula for momentum is p = mv, where p is the momentum, m is the mass of the object, and v is its velocity.
Momentum is a vector quantity because it has both magnitude and direction. The direction of the momentum is the same as the direction of the velocity.
When an object’s speed is doubled, its momentum also doubles because momentum is directly proportional to velocity.
When an object’s speed is doubled, its kinetic energy increases by a factor of four because kinetic energy is directly proportional to the square of the velocity.
No, an object cannot have momentum without having kinetic energy because momentum is the product of mass and velocity, and velocity is a measure of an object’s motion.
Yes, an object’s mass affects both its kinetic energy and momentum. The kinetic energy is directly proportional to the mass, while the momentum is the product of the mass and velocity.
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