Movimiento Proyectil

Projectile Motion

Abstract
The main purpose of this experiment is to determine the projectile motion of an object in two dimension, both X and Y. We will also determine its initial velocity at a specific angle based on the range, height and gravity and compare the experimental ranges with the theoretical ranges. With a height of 1.18m and a gravity of 9.785 m/s2 the initial velocity wascalculated to 4.785 m/s. When comparing the theoretical with the experimental range, a percentage of error of 2.581 was found.

1 Introduction
A projectile is an object or particle which motion is dictated by the force of gravity and whose path is called trajectory. Since gravity is the only force acting on the object, after it is given an initial velocity, the object will free-fall in verticaldirection with a constant velocity at a horizontal direction.

2 Theory
The most important experimental fact about projectile motion in two dimensions is that the horizontal and vertical movements are completely independent of one another. This meaning that the movement in one direction does not have effect on the movement in the other direction.

If an object that is projected horizontally witha velocity (Vox) from some initial height (H), above the floor, the object will travel a horizontal distance (R), during the time it falls a vertical distance (H) (figure 1). Ignoring the effects of air resistance and the rotation of the earth, the trajectory of the projectile in the gravity field of the earth is a curve in the form of a parable (Serway/Vuille, 2010, p.62).

Figure 1 Parabolictrajectory of a particle leaving the origin with a velocity of ῡ0.
Since the velocity in the horizontal direction is constant,
R = V0x t (1)

where t is the time that the object is in flight (which is also the time it takes the object to fall a distance H).

In free fall, the vertical distance moved during a time interval, t, is given by the equation,

Y – Y0 = V0y t - (1/2)gt2(2)

where Y0 is the initial position of the object, g is the acceleration due to gravity (about 9.785 m/sec2 in this experiment), and V0y is the initial velocity of the object in the vertical (y) direction.

In equation (2), positive is taken as the upward direction, and negative direction downward. For objects propelled horizontally, Voy is zero (no component of initial velocity upor down). If the object is initially propelled from a height H above the floor, (Yo = H) then at a later time it hits the floor, and Y = 0. Thus from equation (2) we have,

-H = -(1/2)gt2 (3)

And the time of the flight is,

t = √2H/g (4)

The initial velocity of the projectile can be calculated from equation (1).

2.1 Projectile fired at a given angle (θ)
With a t = 0the projectile leaves its origin with an initial velocity ῡ0. If the velocity vector makes an angle θ0 with the horizontal, where θ0 it’s the angle of projection, at a high H, the range R can be found with kinematics equations. We take equation (1) and clear for velocity and with sin and cos equations we have:

V0x = V0 cosθ0 (5.1)

V0y = V0 sinθ0 (5.2)

where V0x is the initialvelocity (at t = 0) in the x direction and V0y is the initial velocity in the y direction, V0 is the final velocity and θ0 is the angle of projection.

When rewriting equation (2) with equation (5.2) the term Y0 is replaced by the term H, which is the height from the floor to the bottom of the projectile. Then we have,

1/2gt2 – V0senθ t – H = 0 (6)

where V0senθ is the initialvelocity of the projectile.

Since equation (6) is a second order polynomial, t can be found by the quadratic equation,

(7)

Where a = 1/2g , b = V0senθ and c = -H. Then we have,

t = [Vosenθ ± √V02sen2θ + 2gH]/ g (8)

Then we take the result of equation (8) and place it on equation (1) to find the range R.

3 Experimental Procedure
For this experiment we wanted to measure...