A projectile is a body that is
having a two-dimensional motion. A body thrown, making an angle with the
horizontal other than 90° can have a two-dimensional motion. We shall project
the body with a initial velocity for this to happen. This initial velocity can
be resolved into horizontal and vertical components. There is no influence of
acceleration due to gravity along the horizontal direction and hence velocity
component along the horizontal direction always remains constant. As the
gravity acts along the Y direction, the velocity component along the Y
direction keeps changing with respect to time. Here we are going to find out
the effect to velocity of the projectile after a specified time.

Let us consider a body is
projected with the initial velocity making an angle with the horizontal. We can
calculate the horizontal and vertical components of velocities of the body
using the equation of motion. Being these two are perpendicular components to each
other, using the parallelogram law that starts with can find the effective
velocity of the projectile as shown below. It can be noticed that at the
maximum height vertical component of the velocity zero and hence the velocity
of the projectile is equal to the horizontal component of velocity itself. And
hence projectile will have least possible allows state the maximum height but
that is not equal to 0. We can also calculate the change in the moment of the
projectile between the two points of its initial and final journey.

Let us calculate the time after
which initial and final the last is of the projectile are perpendicular to each
other. We can write the velocity of the projectile in vector format as shown
below. Both initial and final velocities of the projectile were returned the
vector formats. We know that if two vectors are perpendicular to each other
their scalar product has to be zero. Hence we can find the scalar product of
the initial and final at the arts of the projectile and we can equate to 0. By
doing this we can identify the time after which initial and final velocity of
the projectile are perpendicular to each other as shown below.

Using the similar sort of concept
we can also calculate the velocity of the projectile at half of its maximum
height. Even in this case the horizontal component of velocity remains
constant. We can find out the vertical component of the velocity at the half of
the maximum height using the third equation of motion of kinematics as shown
below. Anyway these two components are always perpendicular to each other and
by using the parallelogram law sectors we can find the resultant vector as
shown below.

**Problem on velocity of projectile**

A body is projected with a
initial velocity and by making an angle with the horizontal. The body makes 30°
angle with the horizontal after two seconds and then after one second it
reaches the maximum height. What is the angle of projection and what is the
speed of the projectile?

**Solution**

We know that horizontal and
vertical components of elasticity different and we can find the angle between
them using a trigonometrical equation.
By equating this value with the 30° with can get one relation. When the body
reaches the maximum height the angle made by the body with the horizontal is
zero and again by equating with this value with can get one more a question. By
solving these two equations we can get the velocity of the projectile and angle
of the projectile as shown below.

**Problem on projectile motion**

A football is kicked with a
initial velocity of 19.6 m/s with an angle of projection of 45 degree by one
player. On the goal line 67.4 m away from this player and from the direction of
the click, another player start running to meet the ball at the same instant.
What must be his speed to catch the ball before it lands in the goal post ?

**Solution**

We know that as the ball is
projected with a 45° angle it’s angle of projection helps the body to reach its
maximum range and we can calculate the maximum range using its formula as the
39.2 m. And we can also calculate the time taken by the body to reach the
ground using the time of fight formula as shown below. If the other player has
to catch the ball he has to reach the ball before it strikes the ground in the
same time that is equal to time of flight. As he is running on the ground with
a uniform velocity we can use a simple formula that the distance that it has to
cover is equal to product of velocity and time. Hence we can solve the problem
as shown below.

**Expression of range and maximum height in terms of constants of the parabolic equation**

we can express range and maximum
height in terms of constants of the parabola as shown below.

**Related Posts**

**Distance and displacement**

__comparison__
two particles projected in opposite directions horizontally from a given point in space where gravity g is uniform. If u1 and u2 are their initial speeds then what is the time after which their velocities are mutually perpendicular?

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