Projectile Motion - How to find Angle of Projection for same Range #12

This problem of projectile motion finds how you can find 2 angles of projection to get the same range of motion. It makes use of the parabolic equation of motion of a projectile

Imagine a cannon ball fired vertically up from a cannon with a velocity Vo. Also, let us assume there is zero wind condition or the air/ wind is not affecting the motion of the ball

1. It will move vertically up with an initial velocity Vo
2. The velocity of the ball will reduce as it moves up due to the force of gravity acting in downward direction. The velocity will become zero when it reaches maximum height.
3. Say it takes time t
4. The ball will start coming back and its velocity will increase due to the force of gravity acting on it. It will hit the ground in time t again and hit it with same velocity Vo
5. There would be zero vertical displacement since the ball came back where it started from and zero horizontal displacement

Now let us imagine what will happen to the ball if instead of shooting it up vertically, the angle of the cannon is changed so that it is at an angle theta from the ground level while the velocity remains Vo. This then is called a projectile motion
Let us try to examine what is happening to ball as it comes out of the cannon. The moment it starts it has a velocity Vo at an angle theta. This velocity is making the ball move to the vertically up and also horizontally to the right. So let us first see what is happening to the balls movement in the vertical direction

Vertical Movement - The initial velocity of the ball in the vertical direction is Vo Sin (Theta). You will notice that there is force of gravity acting on the ball in vertical direction. Thus the vertical component VoSin (theta) will constantly reduce in the vertical direction till it reduces to zero.
Infact the ball will follow all rules of motion applicable to any body moving in vertical direction under force of gravity. And therefore all equations of motion that we derived for motion in vertical direction would be applicable. So the velocity will reduce in accordance with the following equation

V= Vo –gt till it becomes zero

its displacement in vertical direction will be X = Vi t – 1/2g t(sq)
And all these rules are applicable because the ball is being acted on by gravity. It does not really matter that the ball was thrown up vertically or at an angle theta.

Horizontal Movement - But what happens to the velocity of the ball in the horizontal direction. You would see that gravity is acting on the ball in vertical direction but there is no force acting on the ball in horizontal direction. If there is no force acting in the horizontal direction, there is no acceleration acting on the ball. If there is no acceleration there is no reason for velocity of the ball to change. Hence the velocity of the ball continues to remain the same as the initial velocity in the horizontal direction. It continues to be Vo Cos (theta).

In projectile motion the vertical and horizontal motion are independent of each other. In other words, neither motion affects each other. So when you solve problems in Projectile motion, it would be a good idea to deal with the motion in vertical and horizontal movement separately.

We will first examine the motion in vertical direction first. And I would suggest that since horizontal motion has no affect on the vertical motion you do not even think about horizontal motion for now We know that V= VoSin (theta) – gt.

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