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The relation between kinetic energy and momentum of a body is that both quantities are linked with velocity.
- The kinetic energy of a body is a type of energy that a body possesses due to its motion.
- The momentum of a body is also due to the motion of the body, but it is not a type of energy.
- Momentum is an inherent characteristic of a moving body.
- A simple definition of momentum is mass in motion.
- It indicates that every object with mass m that moves has momentum.
- The amount of momentum an object is given by the product of the mass and velocity of the body.
| Table of Contents |
Key Terms: Kinetic energy, Momentum, Motion, Relation between Kinetic Energy and Momentum, Velocity, Mass, Acceleration
Kinetic Energy
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The kinetic energy of an object is the energy it gains as a result of its motion.
- The kinetic energy of a body is equal to the amount of work required to accelerate a body of mass m from rest to its specified velocity.
- The body maintains its kinetic energy after gaining it during acceleration unless its speed changes.
- The body does the same amount of work while slowing down from its current speed to rest.
- Kinetic energy is a scalar quantity, and its SI unit is Joule or kg m2/s2.
- The dimensional formula of kinetic energy is [M L2 T-2].
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Formula of Kinetic Energy
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The kinetic energy formula is given by
| \(K.E = \frac {1}{2}mv^2\) |
Where
- m is the mass of the body in kg
- v is the velocity in m/s
Momentum
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Momentum is a property of a moving body that the body possesses by virtue of its mass and velocity.
- It is defined as the product of the mass and velocity of the moving body.
- It is a vector quantity since it has both magnitude and direction.
- The SI unit of momentum is Kg m/s.
- The dimension formula of momentum is [M L T-1].
Formula of Momentum
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The momentum formula of a moving object can be given by
| \(p=mv\) |
Where
- p is the momentum
- m is the mass of the body in kg
- v is the velocity in m/s
Relation between Kinetic Energy and Momentum
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The kinetic energy and momentum of a moving object are related as follows.
Consider a body of mass m, moving with velocity v, then the kinetic energy of the body is given by
\(K.E = \frac {1}{2}mv^2\)
Multiply and divide the R.H.S of the above equation by the mass of the body m, and we get
\(K.E = \frac {1}{2}mv^2 \times \frac {m}{m}\)
\(\Rightarrow K.E = \frac {1}{2} \frac {m^2v^2}{m} = \frac {(mv)^2}{2m}\)
The momentum of the body is given by
\(p=mv\)
On substituting the value, we get
| \(K.E = \frac {p^2}{2m}\) |
From the above equation, we can also write
| \(p= \sqrt{2m(K.E)}\) |
The above equations give the relationship between the kinetic energy and momentum of a moving body.
From these equations, we can conclude
| If the kinetic energy (K.E) of the body is constant, then | \(p \propto \sqrt m\) |
| If the momentum (p) of the body is constant, then | \(K.E \propto \frac {1}{m}\) |
| If the mass (m) of the body is constant, then | \(p \propto \sqrt {K.E}\) |
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| Physics Related Concepts | ||
|---|---|---|
| Sound Intensity Formula | Magnetization and Magnetic Intensity | Electrical Energy and Power |
| Power Formula | Light Energy | Light sources |
Things to Remember
- The kinetic energy of a body is by virtue of its motion.
- The formula of kinetic energy is, K.E. = 1/2 (mv2).
- It is a scalar quantity and its SI unit is Joule.
- The momentum of a body is the product of its mass and velocity.
- The formula for momentum is, p = mv
- It is a vector quantity, and its SI unit is kg m/s.
- The relation between kinetic energy and momentum is, K.E = p2/2m
Sample Questions
Ques. Define momentum. (1 Mark)
Ans. The momentum of a body is due to its mass and velocity. It is defined as the product of the mass and velocity of the body.
Ques. What is kinetic energy? (1 Mark)
Ans. The kinetic energy is the energy associated with the moving object. It is referred to as "the energy required by a body to accelerate from rest to stated velocity."
Ques. State the law of conservation of momentum. (2 Marks)
Ans. According to the law of conservation of momentum, the total momentum of two or more bodies acting on one other in an isolated system remains constant until an external force is applied. As a result, momentum cannot be created or destroyed.
Ques. What is the formula for the law of conservation of momentum? (2 Marks)
Ans. The formula for the law of conservation of angular momentum is given by
m1u1 + m2u2 = m1v1 + m2v2
Where
- m1 and m2 are the masses of the bodies
- u1 and u2 are the initial velocities
- v1 and v2 are the final velocities
Ques. The law of conservation of momentum is based on which law of motion? (1 Mark)
Ans. The law of conservation of momentum is based on Newton's third law of motion, which states that every force has an equal and opposite reciprocating force.
Ques. What is the relation between kinetic energy and momentum? (1 Mark)
Ans. The relation between kinetic energy and momentum is given by
K.E. = p2 / 2m or p = √[2m(K.E.)]
Ques. If the Kinetic Energy of an object with a mass of 2 kg is equal to 25 kg m2/s2, then find the momentum of the object. (2 Marks)
Ans. Given
- Kinetic Energy, K.E = 25 kgm2/s2
- Mass of the object = 2kg.
Therefore, by using the formula p = √{2m(K.E.)},
p = √{ 2 . 2 (25)} = √(100) = 10
Therefore, the momentum of the given object, p = 10 kg m/s.
Ques. If the Kinetic of the object is 25 kg m2/s2 and the momentum of the object is 10 kg m/s, then find the velocity of the object. (2 Marks)
Ans. Given
- The kinetic energy of the object, K. E = 25 kg m2/s2
- Momentum = 10 kg m/s.
We know that,
K.E. = 1/2 p . v
Therefore, using the above formula,
25 = 1/2 x 10 . v
⇒ v = 5 m/s.
Therefore, the velocity of the given object is 5 m/s.
Ques. The initial momentum of an object is ‘p’ and the kinetic energy is ‘K’. If the momentum of the object is increased by 50%, find the new value of Kinetic Energy. (2 Marks)
Ans. The initial value of momentum is ‘p’ and the initial value of is ‘K’.
A 50% increase in the value of ‘p’ makes it ‘1.5p’.
Putting this value in the equation
K.E. = 1/2 p . v
We get,
K.E. = 1.25 K
The new value of kinetic energy is 1.25K.
Ques. Derive the relation between Kinetic Energy and momentum of an object with mass ‘m’ and velocity ‘v’. (3 Marks)
Ans. We know that The energy that every substance has when it accelerates is known as kinetic energy, whereas the mass of an item in motion is known as momentum.
And,
K.E. = 1/2 m. v2
And, p = m. v
So we can write Kinetic Energy as:
K.E. = 1/2 (m . v). v
Hence,
K.E. = 1/2 (p. v)
Also, Kinetic Energy can be written as:
K.E. = 1/2 m. v2 . (m / m)
K.E. = 1/2 (m2 . v2) / m
Hence,
K.E. = p2 / 2m or p = √{2m(K.E.)}
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