# Solving Problems based on Newton’s Laws

Laws of Motion describe how objects move under the influence of different types of forces. These forces can be due to any physical phenomenon, but their effect is the same. All the forces change the momentum of the system on which they are acting. Newton gave three laws, these laws describe the interaction between two objects and the forces between them. These laws become the building block for the further theory of mechanics and motion. Let’s look at these concepts and some problems with them.

### Newton’s Laws of Motion

Sir Isaac Newton was an English physicist and mathematician who gave the three laws of motion which form the base for classical mechanics. These laws still keep giving accurate predictions except for bodies traveling with speed comparable to light or the size of an electron. These were the first laws that described the forces acting on the bodies and the motion of the body which is governed by these forces.

**Newton’s First Law: Law of Inertia**

This law states that if a body is at rest or is moving in a straight line with constant speed. It will keep moving in a straight line at constant speed or will remain at rest until it is acted upon by an external force. This property of any object to resist a change in its state is called inertia and thus this law is known **Law of Inertia**.

**Newton’s Second Law: Law of Force and Acceleration**

This law is a quantitative description of the changes that take place when an external force acts on the body. The momentum of the body is defined as the product of the mass and velocity of that body. When a force acts on the body, it brings about changes in the momentum of the body or its direction or both. It is one of the most important laws in the field of classical mechanics. Assuming the mass of the body is “m”, the law is given by,

**F = ma**

Here, F is the force acting on the particle, and “a” denotes the acceleration produced in the body. The direction of acceleration is the same as the direction of motion.

**Newton’s Third Law: Law of Action and Reaction **

The third law of newton states that when two bodies interact with each other, they apply forces to one another which are equal in magnitude and opposite in the direction. This law is also known as action-reaction law. It allows us to explain phenomena such as static equilibrium, where all the forces are balanced, but it also applies to bodies in uniform or accelerated motion. If the net forces acting on the body are equal, the body is said to be in equilibrium.

### Sample Problems

**Question 1: Calculate the momentum of a ball thrown at a speed of 10m/s and weighing 800g. **

**Solution: **

Given: M = 800g and V = 10 m/s

Momentum is given by,

p = MV

Plugging in the values in the formula

p = MV

⇒p = (800)(10)

⇒p = 8000 gm/s

⇒p = 8 × 10

^{3}gm/s

**Question 2: Calculate the momentum of a ball thrown at a speed of 10m/s and weighing 20g. **

**Solution: **

Given: M = 20g and V = 10 m/s

Momentum is given by,

p = MV

Plugging in the values in the formula

p = MV

⇒p = (20)(10)

⇒p = 200 gm/s

⇒p = 2 x 10

^{2}gm/s

**Question 3: A force of 20N is acting on a body of mass 2Kg. Find the acceleration produced. **

**Answer: **

Given:

m = 2Kg

F = 20 N

Acceleration will be given by,

F = ma

Plugging in the values,

F = ma

⇒ 20 = (2)(a)

⇒ 10 m/s

^{2}= a

**Question 4: A force of 100N is acting on a body of mass 5Kg. Find the acceleration produced. **

**Answer: **

Given:

m = 5Kg

F = 100 N

Acceleration will be given by,

F = ma

Plugging in the values,

F = ma

⇒ 100 = (5)(a)

⇒ 20 m/s

^{2}= a

**Question 5: A body of 2Kg is moving at a velocity of 50m/s. A force starts acting on it and the velocity becomes 20m/s in a time of 5 seconds. Find the force applied on the body. **

**Answer: **

Given:

m = 5Kg

v

_{i}= 50 m/s,v

_{f}= 20 m/s.t = 5 s

Force is defined as rate of change of momentum.

F = m(v

_{f}– v_{i})/t⇒ F = (5)(50 – 20)/(5)

⇒ F = 30N

**Question 6: A body of 10Kg is moving at a velocity of 100m/s. A force starts acting on it and the velocity becomes 20m/s in a time of 10 seconds. Find the force applied to the body. **

**Answer: **

Given:

m = 10Kg

v

_{i}= 100 m/s,v

_{f}= 20 m/s.t = 10 s

Force is defined as rate of change of momentum.

F = m(v

_{f}– v_{i})/t⇒ F = (10)(80 – 20)/(10)

⇒ F = 80N

**Question 7: The momentum of the body is given by the equation below, **

**p(t) = 3t ^{2} + 4t + 5**

**Find the force acting on the body at t = 5. **

**Answer: **

The force is given as rate of change of momentum,

F = dp/dt

Given:

p(t) = 3t

^{2}+ 4t + 5⇒ F = 6t + 4

At t = 5

F = 6(5) + 4

⇒ F = 34 N

**Question 8: The momentum of the body is given by the equation below, **

**p(t) = e ^{t} + t^{2} + 20**

**Find the force acting on the body at t = 0. **

**Answer: **

The force is given as rate of change of momentum,

F = dp/dt

Given:

p(t) =

e^{t}+ t^{2}+ 20⇒ F =

e^{t}+ 2tAt t = 0

F = 1

⇒ F = 1 N