PROJECTILE MOTION
INTRODUCTION
In the previous chapters, we have been discussing the motion of bodies, either in
horizontal or vertical directions. But we see that whenever a particle is projected
upwards at a certain angle (but not vertical), we find that the particle traces some path
in the air and falls on the ground at a point, other than the point of projection. If we study
the motion of the particle, we find that the velocity, with which the particle was projected,
has two components namely vertical and horizontal.
The function of the vertical component is to project the body vertically upwards,
and that of the horizontal is to move the body horizontally in its direction.
The combined effect of both the components is to move the particle along a parabolic
path. A particle, moving under the combined effect of vertical and horizontal forces, is
called a projectile. It may be noted that the vertical component of the motion is always
subjected to gravitational acceleration, whereas the horizontal component remains
constant.
IMPORTANT TERMS
1) Trajectory: The path, traced by a projectile in the space.
2) Velocity of projection: The velocity, with which a projectile is thrown.
3) Angle of projection: The angle, with the horizontal at which a projectile is thrown
4) Time of flight: The total time taken by the projectile to reach the maximum height
and to return back to the ground.
5) Range: The distance, between the point of projection and the point where
projectile strikes the ground.
MOTION OF A PROJECTILE
Consider a particle projected upwards from a point O at an angle α, with the horizontal,
with an initial velocity U m/s as shown below.
Now resolving this velocity into its vertical and horizontal components,
V = u sinα and H = u cosα
We know that the vertical component (u sinα) is subjected to retardation due to gravity.
The particle will reach maximum height, when the vertical component becomes zero.
After this the particle will come down, due to gravity and this motion will be subjected to
acceleration due to gravity
The horizontal component (u cosα) will remain constant, since there is no acceleration
or retardation. The combined effect of the horizontal and vertical components will move
the particle along some path in the air and then the particle falls on the ground at some
point A.
EQUATION OF THE TIME OF FLIGHT OF A PROJECTILE ON A HORIZONTAL
PLANE
Time of flight is the time for which the projectile has remained in the air
2𝑢𝑠𝑖𝑛𝑎
t=
𝑔
Problem 1
A projectile is fired upwards at an angle of 30 degrees with a velocity of 40m/s.
Calculate the time taken by the projectile to reach the ground after the instant of firing.
Solution:
2𝑢𝑠𝑖𝑛𝑎 2×40𝑠𝑖𝑛30 80×0.5
t= = = = 4.08s
𝑔 9.81 9.81
EQUATION OF THE HORIZONTAL RANGE OF A PROJECTILE ON A HORIZONTAL
PLANE
Horizontal range is the horizontal distance between the point of projection and the point,
where the projectile returns back to earth.
𝑈 2 sin 2𝑎
R=
𝑔
Problem 2
A ball is projected upwards with a velocity of 15m/s at an angle of 25 degrees with the
horizontal. What is the horizontal range of the ball?
𝑈 2 sin 2𝑎 (15)2 sin(2𝑋25) 225 sin 50 225 x 0.766
R= = = = = 17.6m
𝑔 9.81 9.81 9.81
EQUATION FOR THE MAXIMUM HEIGHT OF A PROJECTILE ON A
HORIZONTAL PLANE
Maximum height of a projectile is the vertical distance between the ground and the point
in the air the particles final velocity reaches zero
𝑈 2 𝑋𝑠𝑖𝑛2 𝑎
H=
2𝑔
Problem 3
A bullet is fired with a velocity of 100m/s at an angle of 45 degrees with the horizontal.
How high is the bullet going to rise?
𝑈 2 𝑋𝑠𝑖𝑛2 𝑎 (100)2 𝑋𝑠𝑖𝑛2 45 10000𝑋(0.707)2
H= = = = 255.1𝑚
2𝑔 2(9.81) 19.6
