Motion in Two Dimension
Two-dimensional motion is sometimes called "projectile motion" which encompasses objects flying through space under the influence of gravity. Baseballs, cannon balls, basketballs moving through space are all examples of projectile motion.
Projectile Motion
Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only.
The only force of significance that acts on the object is gravity, which acts downward to cause a downward acceleration. There are no horizontal forces needed to maintain the horizontal motion – consistent with the concept of inertia.
Figure: Projectile Motion
Projectile and Trajectory
A body projected with a uniform velocity at an angle with the horizontal in the vertical plane of the earth, is called a projectile. The path traversed by the projectile is called its trajectory.
02
Components of Velocity
Consider, a projectile is launched at an angle to the Earth's surface. When this occurs, the velocity in the vertical direction is no longer equal to zero. There is velocity in both the horizontal and vertical directions, since the projectile moves up and down as it moves.
Mathematically
Horizontal Component
Vertical ComponentMaximum Height
The maximum height of the object is the highest vertical position along its trajectory.
The maximum height of the projectile depends on the initial velocity , the launch angle , and the acceleration due to gravity.
We know the equation of motion due to gravity is
Let, the maximum height reached by the projectile be H where final velocity is zero ( ). So, the equation becomes
But, the vertical component of the velocity is
From equation number (1)
When θ = 900, the height will be maximum. Then, this equation will be
Time to Reach Maximum Height
We know the equation of motion due to gravity is
But, the vertical component of the velocity is and at the maximum height, the final velocity is 0.
From equation number (1)
Time of Flight
Let the time of flight be T. Now, we know
The time of ascend to the maximum height = time of descend to the ground So, the Total Time of flight can be written as
Horizontal Range
The distance travelled along the horizontal direction in the time of flight is called the horizontal range.
This is denoted by R.
Mathematically
R = horizontal component of the initial velocity × time of flight
It is evident that R will be maximum, when
For that case, the equation becomes
It is clear that, if an object is thrown at an angle with the horizontal direction, the horizontal range will be maximum.
General Equation of Parabola
Consider, a projectile started its journey from a point having initial velocity making and angle with the horizontal axis .
When time,
The horizontal component of the initial velocity
The vertical component of the initial velocity
From the equation of motion
Acceleration works only in the vertical direction which is gravitational acceleration.
When, time .
The projectile reaches at point whose coordinate is and the velocity is . So, the displacement of the projectile along the X axis is OA (OA = x). We can write
Now, the vertical component of the velocity
The vertical displacement at time
Using equation number (2) in equation number (3)
In this equation are constants. Let and
as constants.
Finally, equation number (4) becomes
This is an equation of a parabola. Hence, the path of motion (called trajectory) of a projectile is parabola.
