NEWTON'S LAWS OF MOTION

Mechanics is a branch of physics that deals with motion of objects/bodies in time and space. There are universal laws that govern the motion of bodies in space and time. Newton's laws are one of those kinds.

An object in motion is associated with momentum. If we know the initial position of the object, we can possibly determine its trajectory with the help of a set of equations. The position of an object is always measured with respect to the observer.

For all practical purposes, we fix a reference system and then we describe the motion of the object concerned. Here, we would be using Cartesian coordinate system to establish the relation between space and time. Elementary ideas of calculus are needed to solve this type of problems.

STATE OF AN OBJECT:

REST: A body is said to be at rest when it does not change its position with time. For example, a book kept on a table.

MOTION: A body is said to be in motion when it changes with position with time. For example, a moving car.

However, the notion of rest and motion is relative since it depends on the position of observer.

A book kept on a table might seem to be moving for an observer who is on a train. Hence, frame of reference plays an important role for complete description of an object.

In a nutshell, we can say that motion is a property of object under study and the observer.

Newton's laws of motion are a set of equations that determine the trajectory of a point particle over time. It is directly applicable only when the object is moving with a constant acceleration.

Suppose, the object has acceleration ‘a’ at some time t₀, then this acceleration ‘a’ remains a constant over time. Acceleration by definition is the rate of change of velocity; hence constant acceleration implies that this rate of change remains a constant over time.

From differential calculus, we know

The above three equations are popularly called newton's equation of motion. They are useful in solving in motion in 1D with constant acceleration.

X denotes the position of an object at any time.

U denotes the initial velocity, a vector quantity.

V is the final velocity.

T is the time.

FREELY FALLING BODIES:

It is a well-known example of motion in a straight line when air friction and other dissipative forces are not taken into account.

If the body is not far from the surface of earth, then acceleration ‘a’ is replaced by acceleration due to gravity ‘g’ in the above equations. Since acceleration is a vector quantity, it is always associated with a direction. Here ‘g’ is in downward direction and its value is 9.8m/s² or 32ft/s².

Before solving any problem, Fix your coordinate system and mark all the vector quantities accordingly.

Equations in this case take the form:

MOTION IN A PLANE

When a particle is free to move in a plane, it has 2 degrees of freedom. Hence, we need two position coordinates to explicitly state the position of particle. At any

Now that we have separated all the parameters in two coordinates, we can write equations of motion along X and along y separately.

APPLICATION OF LAWS OF MOTION:

PROJECTILE MOTION:

It is a motion in plane where acceleration is constant. When a particle is thrown at some angle near surface of earth, the path traced by the particle is a curved one. This particle is called projectile and its motion: projectile motion.

From the above set of equations, we can determine time of flight, maximum height reached and range.

Let T=time flight; the time after which projectile lands on ground

Y coordinate should be zero in this case, substituting the required condition, we get:

There may be different conditions imposed on various problems based on projectile motion. However, underlying principle remains the same. Apply newton's laws of motion along X-axis and Y-axis using the definite conditions given.

Note: Newton's laws are only directly applicable when acceleration is constant. For variable acceleration cases, we can not apply newton's laws of motion.