A body is said to be in
translatory motion if all the particles of the body are having a similar kind
of displacement and velocities. If this body is moving only along one
direction, it is called one-dimensional motion. If the body is thrown with an angle,
then it will have motion both along the x-axis and y-axis simultaneously and
this kind of the motion is called two-dimensional motion. It is also called

**projectile motion**.
If the body projected with an
angle other than 90° with the horizontal then, it is called projectile. We can
study displacement and velocity of this kind of motion simultaneously on both X
and Y axis. We can also find out what is the maximum horizontal distance that
the body can travel. This maximum horizontal distance that a projectile travels
is called the range. The time taken by the projectile to reach the ground from
the point of projection is called Time of flight. The maximum vertical distance
that it can reach is called maximum height. Here we are going to derive the
equations for all this values.We are ignoring all the impact of air friction on the motion of the body during this study for simplicity.

Let us consider that you are
throwing a stone into the air but not vertically up. On this stone now there
are two forces acting simultaneously. One force is the force that you have
applied and another one is the gravitational force which is always acting in
the vertically downward direction. The resultant motion of the body is due to
these two forces. The gravitational force acting on the body is constant and
because of its influence it finally comes down and then reaches the ground. Until
the body reaches the maximum height, it is moving against the gravity and hence
it’s velocity keep on decreasing. From the maximum height point on words the
body is coming in the downward direction and gravity is also pulling it in the
downward direction. Hence it’s velocity keep on increasing further.

Let us consider a body having a
mass stone with an angle to the
horizontal with a initial velocity. The velocity of the body is in between X
and Y axis and hence it can be resolved into components. The horizontal
component of the velocity at along the x-axis and the vertical velocity is
along y-axis. As there is no gravitational effect along the x-axis, this
horizontal component of velocity always remains constant. But the velocity
along the y-axis keep on changing with respect to time as gravity influences also
changes.

Using the equations of motion we
can write displacement of the particle along both x-axis and y-axis and by
substituting the value of the time from the x-axis equation on the y-axis
equation, we can prove that part of this body is projectile. It is as shown
below.

The final equation of the
displacement along y-axis represents a parabola according to mathematics. It is
probably learned in the school level that the equation says that the body is
having a simultaneous motion along both x-axis and y-axis and it takes a

**parabolic path**.
We can further derive the
equation for a

**time of flight**. At the end of the time of the flight as the body is coming back to the ground, its displacement along y-axis is equal to**0**. By equating the equation of the displacement to 0 with can get the equation for the time of flight as shown below. It can be also further noted that this time of flight is the sum of time of ascent and the time of dissent. Time of ascent is the time taken for the body to reach the maximum height and time of dissent is the further time taken by it to reach the ground.
Once if you know the equation of
the time of flight, by putting dirty equation value in the displacement of the
body along x-axis, we can calculate the total distance travelled by the body
along x-axis. This total distance travelled by the body on the horizontal axis
is called

**horizontal range**.
It can be further notice that
horizontal range will be maximum if the angle of projection is 45° . If we have
noticed any of the athlete throwing a discus throw ball, he preferred to throw
the ball by making an angle of 45° with the horizontal so that it can go for
the maximum horizontal distance and hence he will be the winner.

It can also be proved that

**range is a projectile is equal for two angles of projection**.
The horizontal component of the
velocity always remains constant, as there is no gravitational impact in the
direction. The vertical component of the velocity keeps on decreasing and by
the time the body reaches the maximum height it will become zero. Hence the
projectile at the maximum high it is having only horizontal component of
velocity and it is the least possible velocity that the projectile can have at
any point of the journey. Taking this into consideration and by substituting
the value any question of motion we can derive the equation for the

**maximum height**attained by the projectile as shown below.

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