Nikita starts to play a game of throwing a ball into a net. She says to herself that she is going to throw the ball exactly into the net; not only that, but also she makes it a point to not hit the ceiling when she throws the ball. In order to make the ball drop into the net, she tries multiple times; and she notices that the way she swung her hand made a difference; in other words, the angle at which the ball is being thrown matters a lot; for instance, if it was less than , then the ball falls before the net and misses the ceiling. Not to mention, she needed to throw the ball with certain minimum speed or else she missed the net. Also, she notices the ball follows a parabolic path. So how exactly is the angle at which an object being thrown is going to decide what path it is going to take, how far it is going to hit the ceiling or the ground, and how long it is going to take to do so? Classical mechanics calls such parabolic motions “ projectile motions”. Volleyball, cannon balls, stone thrown in a river all come under the category of projectile motion. So let’s explore projectile motion in detail!
Table of contents
A projectile is an object that is thrown into space and the force acting on it is only due to gravity or one single net constant force. The path followed by a projectile is termed its trajectory.
Javelin throw, throwing a basketball, firing a cannon, an arrow shot from a bow, are all examples of projectile motion.
Consider the following example where a ball is launched from the point with a velocity and an angle with the horizontal. At point , it has a horizontal velocity and vertical velocity . At point it is said to have reached its maximum height; where only the horizontal component will remain. When it hits point its vertical component becomes negative. The horizontal distance is called the range of the projectile.
Let and indicate horizontal and vertical velocities after time
The ball undergoes acceleration only in the vertical direction, hence and
( acceleration due to gravity).
Then according to first equation of motion,
The net velocity after time
1) Time of flight
The total time taken (from point O to point B) is called the time of flight .
Let indicate the time taken for the ball to go from O to A.Then,
(at point A)
,
2)Horizontal Range
Horizontal range( Horizontal component of velocity, Time of flight
(
When
∴ When the ball is projected at it reaches maximum horizontal distance.
Let be the same horizontal range when two projectiles are projected at two angles and with the same speed Then is given by,
I.e. when the angles of projection are complementary to each other, they both have the same range.
3)Maximum height reached
Applying the second equation of motion, vertical height can be calculated as,
Substituting in the above equation, we get
I.e. when the ball is projected at an angle of the range is four times the height reached.
4)Equation for trajectory of a projectile
The horizontal and vertical velocities of a projectile are
Eliminating we get
indicates the range.
Q. A cricket ball is launched at a speed of in a direction making an angle of upwards with the horizontal.
1) What is the maximum height reached by the object?
2) What is the ball’s total flight time?
3) What is the ball’s horizontal range ?
1)
2)Total time of flight
3) Horizontal range
Q. The equation for the trajectory of a projectile is given by Find the angle of projection and range.
A.
Comparing with , we get
Q. The ceiling of a room is high. Find the angle at which a ball may be thrown with a speed of , so that it travels without hitting the ceiling of the room.?
A.
The height of the ceiling is Equating this to maximum height
Q.A particle is projected at an angle of with the horizontal and the other particle is projected at the same time at an angle of The respective heights attained by the particles are and Find the ratio of .
(a). (b) (c) (d)
Height attained
Q. What are the 4 factors that affect projectile motion?
A. Gravity, air resistance, speed of release, angle of release are the four factors that affect projectile motion.
Q. What are the 3 main parameters of projectile motion?
A. Maximum height, Range and Time of flight are the 3 main parameters of projectile motion.
Q. The value of the vertical component of velocity keeps on changing in a projectile. Why?
A.
Acceleration due to gravity acting downwards causes the value of vertical velocity to change with time.
Q. Write the 3 types of projectile motion.
A.
(i) Oblique projectile motion.
(ii) Horizontal projectile motion.
(iii) Projectile motion on an inclined plane.