Sunday, October 11, 2015

Projectile Motion Range,Time of Flight and Maximum Height Equations

A body is said to be in translatory motion if all the particles of the body are having a similar kind of displacement and velocities. If this body is moving only along one direction, it is called one-dimensional motion. If the body is thrown with an angle, then it will have motion both along the x-axis and y-axis simultaneously and this kind of the motion is called two-dimensional motion. It is also called projectile motion.

If the body projected with an angle other than 90° with the horizontal then, it is called projectile. We can study displacement and velocity of this kind of motion simultaneously on both X and Y axis. We can also find out what is the maximum horizontal distance that the body can travel. This maximum horizontal distance that a projectile travels is called the range. The time taken by the projectile to reach the ground from the point of projection is called Time of flight. The maximum vertical distance that it can reach is called maximum height. Here we are going to derive the equations for all this values.We are ignoring all the impact of air friction on the motion of the body during this study for simplicity.

Let us consider that you are throwing a stone into the air but not vertically up. On this stone now there are two forces acting simultaneously. One force is the force that you have applied and another one is the gravitational force which is always acting in the vertically downward direction. The resultant motion of the body is due to these two forces. The gravitational force acting on the body is constant and because of its influence it finally comes down and then reaches the ground. Until the body reaches the maximum height, it is moving against the gravity and hence it’s velocity keep on decreasing. From the maximum height point on words the body is coming in the downward direction and gravity is also pulling it in the downward direction. Hence it’s velocity keep on increasing further.

Let us consider a body having a mass  stone with an angle to the horizontal with a initial velocity. The velocity of the body is in between X and Y axis and hence it can be resolved into components. The horizontal component of the velocity at along the x-axis and the vertical velocity is along y-axis. As there is no gravitational effect along the x-axis, this horizontal component of velocity always remains constant. But the velocity along the y-axis keep on changing with respect to time as gravity influences also changes.

Using the equations of motion we can write displacement of the particle along both x-axis and y-axis and by substituting the value of the time from the x-axis equation on the y-axis equation, we can prove that part of this body is projectile. It is as shown below.

The final equation of the displacement along y-axis represents a parabola according to mathematics. It is probably learned in the school level that the equation says that the body is having a simultaneous motion along both x-axis and y-axis and it takes a parabolic path.

We can further derive the equation for a time of flight. At the end of the time of the flight as the body is coming back to the ground, its displacement along y-axis is equal to 0. By equating the equation of the displacement to 0 with can get the equation for the time of flight as shown below. It can be also further noted that this time of flight is the sum of time of ascent and the time of dissent. Time of ascent is the time taken for the body to reach the maximum height and time of dissent is the further time taken by it to reach the ground.

Once if you know the equation of the time of flight, by putting dirty equation value in the displacement of the body along x-axis, we can calculate the total distance travelled by the body along x-axis. This total distance travelled by the body on the horizontal axis is called horizontal range.

It can be further notice that horizontal range will be maximum if the angle of projection is 45° . If we have noticed any of the athlete throwing a discus throw ball, he preferred to throw the ball by making an angle of 45° with the horizontal so that it can go for the maximum horizontal distance and hence he will be the winner.

It can also be proved that range is a projectile is equal for two angles of projection.

The horizontal component of the velocity always remains constant, as there is no gravitational impact in the direction. The vertical component of the velocity keeps on decreasing and by the time the body reaches the maximum height it will become zero. Hence the projectile at the maximum high it is having only horizontal component of velocity and it is the least possible velocity that the projectile can have at any point of the journey. Taking this into consideration and by substituting the value any question of motion we can derive the equation for the maximum height attained by the projectile as shown below.

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