## Friday, October 16, 2015

### Horizontal Projectile, Applications Problems with Solutions

A body projected horizontally from a certain height with an initial horizontal velocity can be called as a horizontal projectile. Its initial velocity along the vertical direction is zero and it possess only horizontal velocity at the beginning. As the time progresses, due to the impact of the gravity, it acquires the vertical component of velocity also. It can be shown that path taken by this body is parabola using the equations of motion of kinematics. We can write the equations for the displacement along x-axis and y-axis and further substituting the value of the time from the x-axis equation in the y-axis equation, it can be proved that its path is parabola.

At any instant of the path, velocity of the projectile is tangential to its path. It has both horizontal and vertical components and the angle made by the velocity vector with the horizontal can be calculated as shown below. We can also calculate the effective velocity of the projectile at that instant.

Application

Let us consider a body projected horizontally from the top of a tower. The line joining the point of projection and the striking point of ground makes an angle of 45° with the horizontal.  What is the displacement of the body?

Solution

As the angle made at the horizontally is 45 degree, according to trigonometry, its horizontal and vertical distances are equal. Being the displacement is a vector quantity with can calculate its resultant value using the parallelogram law of the vectors as shown below.

Application

Let us consider a ball is thrown horizontally from a staircase as shown with a initial velocity. We shall calculate how many steps it travels before it strikes the ground?

Solution

The ball is having only horizontal component of velocity and has no initial vertical component of velocity.

We can use equations of motion to find the displacement along x-axis and y-axis. There is no acceleration due to gravity impact on the x-axis and there is gravity acting along the y-axis. The total horizontal distance travelled by the body is equal to the multiplication of the number of the steps and the breadth of each step. Similarly the total vertical distance travelled by the body is equal to the multiplication of number of steps with the height of each step.

By substituting this values in the above equation is can solve the problem as shown below.

Application:

Two bodies are thrown horizontally with a two different initial velocities in mutually opposite directions from the same height. What is the time after which velocity is of the two bodies are perpendicular to each other ?

Solution:

Velocity is a physical quantity that is having both magnitude and direction and it has to be treated like a vector quantity. When two vectors are perpendicular to each other their scalar product becomes zero. Taking this into consideration and by writing the equations of velocity is for the two bodies after a specific time and equating their product to 0 with can solve the problem as shown below.

Problem

An object is thrown horizontally from a point it hits the ground at some another point. The line of sight from these two points makes an angle 60° with the horizontal. If acceleration due to gravity is 10 m/s Squire and time of flight is known what is the velocity of projection?

Solution

by writing the equations were displacement along x-axis and y-axis and by using a trigonometrical definition we can solve the problem as shown below.

Problem

Two paper screens are separated by a distance of hundred meter. A bullet fired through them. The hole made in the second screen is 10 cm below the hole made in the first screen. What is its initial velocity before it strikes the first screen?

Solution

Initially the bullet is having only horizontal velocity and its vertical component of velocity equal to 0. There is no impact of acceleration due to gravity along the x-axis and it is acting along the y-axis. Taking this things into consideration we can equate the distance between the two poles along the vertical direction is equal to the vertical distance travelled by the body during the journey time. We can also calculate the time taken by the body for this to happen using the equation of the displacement along the x-axis. By solving these equations together we can calculate the initial velocity of the projection as shown below.

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