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Mathematical Formulation of the Second Law of Motion is needed when we notice some cases like when the magnitude of the force required to push two blocks of wood is kept the same, and one of the blocks is taken lighter than the other, their position will change at different speeds as the lighter block will move faster than the heavier ones. So, the acceleration of an object is gained not only by the factor of force but also a factor of the mass of the object. However, a small bullet when kept stationary doesn’t harm the object, but when it is fired with higher velocity, it can certainly bring harm. So, it can be said that both masses, velocity affect the impact of an object. These cases are what the second law of motion is used to validate. In this article, we will see the mathematical formulation of second law of motion.
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Key Terms: Second Law of Motion, Motion, Laws of Motion, Newton, Acceleration, Mass, Velocity
Second Law of Motion
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Newton’s Second Law of Motion is formally explained through the following sentence:
The acceleration produced by a net force on an object, that is directly proportional to the net force’s magnitude, and is inversely proportional to the object's mass but in the direction of the net force.
Newton’s Second Law of Motion
Notations Used for the Formula
The following notations have been used in the formula for Second Law of Motion.
F is used for the force.
k is used for showing the proportionality constant.
a is used for acceleration.
The SI units used for acceleration and mass are m.s-2 and kg respectively.
So, a unit of force = k x (1 kg) x (1 ms-2)
Read about: Difference between Acceleration and Velocity
Mathematical Formulation of Second Law of Motion
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To Formulate the Second Law of Motion, firstly, let us consider an object with a mass of ‘m’, that is moving with an initial velocity ‘u’ along a straight line. So, After time t, with the help of a constant acceleration ‘a’, the final velocity is ‘v’. So we notice, the initial momentum, (p1) = m*u.
And The final momentum, (p2) = m*v
The change in momentum is written as p2 - p1 = (m*v) - (m*u)
= m *(v-u)
Momentum’s rate of change with respect to time is directly proportional to the force applied. So The applied force, (F) ∝ (m * (v-u))/t
Or, F ∝ m * a
as acceleration (a) = velocity's rate of change with respect to time.
F = k * m * a
This is the formula for the second law of motion.
Read about: Force and Motion
Real-Life Examples of Second Law of Motion
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Some examples or applications of second law of motion include:
- When pushing a truck and a car with the same force. The truck has a larger mass, so it will gain less acceleration than a car that has a smaller mass.
- While riding a bicycle, the bike has mass, and the rider uses force to push the paddles with his/her leg.
- A fielder catches a cricket ball by pulling his/her hand back, to delay the momentum of the ball that is moving at a high speed. If it is not then the ball will exert a high force on the player’s hand and cause a serious injury.
- Athletic sports like high jumps, long jumps use a cushioned bed or bed of sand at the place where the sportsperson lands. So that after jumping on it, his/her momentum slowly becomes zero, and this prevents injury.
Application of Newton’s Second Law of Motion
Things To Remember
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- Acceleration of an object is gained not only by the factor of force but also a factor of the mass of the object.
- A small bullet when kept stationary doesn’t harm the object, but when it is fired with higher velocity, it can certainly bring harm.
- So, it can be said that both masses, velocity affect the impact of an object.
- The acceleration produced by a net force on an object, that is directly proportional to the net force’s magnitude and is in the direction of the net force while being inversely proportional to the mass of the object.
- The SI units used for acceleration and mass are m.s-2 and kg respectively. So, a unit of force = k x (1 kg) x (1 ms-2)
Sample Questions
Ques. Find the force needed to speed up a car of mass 1000kg, with a rate of 5ms–2. (5 marks)
Ans. According to question:
Acceleration (a) = 5m/s2
While Mass (m) = 1000 kg
We know that,
F = m × a
= 1000 kg × 5 m/s2
= 5000 kg m/s-2
Therefore, required Force
= 5000 m/s2 or 5000 N
Ques. What force will be required to speed up a moving object of mass 50 kg at a rate of 2ms-2. (5 marks)
Ans. According to the question;
Acceleration (a) = 2ms-2
While Mass (m) = 50 kg.
We know that,
F = m × a
= 50 kg × 2 m/s2
= 100kg m/s2
Therefore, the required force
= 100 m/s2 or 100 N.
Ques. What is the force needed to accelerate a vehicle of mass 100kg to 3m/s2. (5 marks)
Ans. According to the question:
Acceleration (a) = 3 m/s2
While Mass (m) = 100 kg.
We know that,
F = m × a
= 100 kg × 3 m/s2
= 300 kg m/s2
Therefore, required Force
= 300m/s2 or 300 N.
Ques. Find the mass of the object that requires 10N of force for an acceleration of 2ms-2. (3 marks)
Ans. Given:
Acceleration (a) = 2m/s2,
Force (F) = 10N,
Then find Mass (m).
We know that,
F = m × a
⇒10N = m × 2m/s2
⇒m = 10/2 kg = 5 kg
Hence the object's mass is 5 kg.
Ques. Find the weight of the object that needs 1000N force to accelerate at 5ms-2. (3 marks)
Ans. Given: Acceleration (a) = 2m/s2,
And Force (F) = 1000N,
Find Mass (m).
We know that,
F = m×a
⇒1000N = m × 5m/s2
⇒m = 1000/5 kg = 200 kg
Therefore the object's mass is 200 kg.
Ques. Find the mass of the vehicle that requires 50000 N of force to accelerate at 10ms-2. (3 marks)
Ans. According to the question,
Acceleration (a) = 10 m/s2,
Force (F) = 50000N,
therefore, Mass (m) = ?
We know that,
F = m × a
⇒ 50000 N = m × 10 m/s2
⇒ m = 50000/10 kg
= 5000 kg
Therefore, the vehicle's mass is 5000 kg
Ques. Find the acceleration of the vehicle with 1000 kg of mass, when it is pushed by a force of 5000N. (5 marks)
Ans. According to question:
Mass (m) = 1000 kg
And Force (F) = 5000N
As we know,
Force = Mass times Acceleration
= F = m x a
∴ 5000 N = 1000 kg × a
⇒ a = 5000N/1000 kg
⇒ 5000 kg ms-2/1000 kg
= 5m/s-2
Thus acceleration of the vehicle = 5m/s2
Ques. Find the acceleration of an object of mass of 2000g After applying a force of 1000 N on it. (5 marks)
Ans. We know that,
Mass (m) = 2000 kg
And Force (F) = 1000N
Then Find Acceleration (a).
We know that
Force = Mass x Acceleration or
F = m x a
∴ 1000 N = 2000 kg × a
⇒ a = 1000N/2000 kg
⇒1000 kg ms-2/2000 kg
= 0.5m/s-2.
Thus acceleration of the vehicle = 0.5m/s2.
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