The oblique parabolic shot It is a particular case of the movement of free fall in which the initial velocity of the projectile forms a certain angle with the horizontal, resulting in a parabolic trajectory.

Free fall is a case of movement with constant acceleration, in which the acceleration is that of gravity, which always points vertically downwards and has a magnitude of 9.8 m/s^2. It does not depend on the mass of the projectile, as Galileo Galilei demonstrated in 1604.

If the initial velocity of the projectile is vertical, the free fall has a straight and vertical trajectory, but if the initial velocity is oblique then the free fall trajectory is a parabolic curve, a fact also demonstrated by Galileo.

Examples of parabolic motion are the path followed by a baseball, the bullet fired from a cannon, and the stream of water coming out of a hose.

Figure 1 shows an oblique parabolic shot of 10 m/s with an angle of 60º. The scale is in meters and the successive positions of P are taken with a difference of 0.1 s starting from the initial instant 0 seconds.

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**formulas**

The motion of a particle is fully described if its position, velocity, and acceleration as a function of time are known.

The parabolic movement resulting from an oblique shot is the superposition of a horizontal movement at constant velocity, plus a vertical movement with constant acceleration equal to the acceleration of gravity.

The formulas that apply to the oblique parabolic throw are the one that corresponds to a movement with constant acceleration a = gnote that bold type has been used to indicate that acceleration is a vector quantity.

**position and velocity **

In a movement with constant acceleration, the position depends mathematically on the time in quadratic form.

If we denote r

r

The bold in the previous expression indicates that it is a vector equation.

Velocity as a function of time is obtained by taking the derivative with respect to t of the position and the result is:

v

And to obtain acceleration as a function of time, take the derivative of velocity with respect to you resulting:

**to**

*When time is not available, there is a relationship between velocity and position, which is given by:*

*v2 =veither2 – 2 g (and – i )*

**equations**

**equations**

*Next we will find the equations that apply to an oblique parabolic throw in Cartesian form.*

*The movement begins instantly t=0 with starting position (xo, me) and velocity of magnitude veither and angle θthat is to say that the initial velocity vector is ( veither cosθ, veither sinθ ). The movement is accelerating *

*g = (0, -g).*

**parametric equations**

**parametric equations**

*If the vector formula that gives the position as a function of time is applied and components are grouped and equalized, then the equations that give the coordinates of the position at any instant of time t will be obtained.*

*x*

*y*

*Similarly, we have the equations for the velocity components as a function of time.*

*vx*

*vand*

*Where: vox = vo cosθ ; voy =veither sinθ*

**trajectory equation**

**trajectory equation**

*y = Ax^2 + Bx + C*

*A = -g/(2 vox^2)*

*B = ( voy/vox +gxeither/vox^2)*

*C = (andeither –voy xeither /vox)*

**examples**

**examples**

**Example 1**

**Example 1**

*Answer the following questions:*

*a) Why is the effect of air friction usually neglected in parabolic draft problems?*

*b) Does the shape of the object have any importance in the parabolic shot?*

**Answers**

**Answers**

*a) For the movement of a projectile to be parabolic, it is important that the force of air friction be much less than the weight of the object that is thrown. *

*If a ball made of cork or some light material is thrown, the friction force is comparable to the weight and its trajectory cannot approach a parabola. *

*On the contrary, if it is a heavy object like a stone, the friction force is negligible compared to the weight of the stone and its trajectory does approach a parabola.*

*b) The shape of the thrown object is also relevant. If a sheet of paper is thrown in the shape of a small plane, its movement will not be free fall or parabolic, since the shape favors air resistance.*

*On the other hand, if the same sheet of paper is compacted into a ball, the resulting movement is very similar to a parabola.*

**Example 2**

**Example 2**

*A projectile is launched from the horizontal ground with a speed of 10 m/s and an angle of 60º. These are the same data with which Figure 1 was made. With these data, find:*

*a) Instant at which it reaches the maximum height.*

*b) The maximum height.*

*c) The speed at the maximum height.*

*d) The position and velocity at 1.6 s.*

*e) The instant at which it hits the ground again.*

*f) The horizontal reach.*

**Solution to)**

**Solution to)**

*The vertical velocity as a function of time is*

*vy*

*At the moment the maximum height is reached, the vertical velocity is zero for an instant. *

*8.66 – 9.8 t = 0 ⇒ t = 0.88 s.*

**Solution b)**

**Solution b)**

*The maximum height is given by the coordinate and for the instant at which that height is reached:*

*y(0.88s) = i + go t -½ gt^2 = 0 + 8.66*0.88-½ 9.8 0.88^2 = *

*3.83m*

*Therefore the maximum height is 3.83 m.*

**Solution c)**

**Solution c)**

*The velocity at maximum height is horizontal:*

*vx*

**Solution d)**

**Solution d)**

*The position at 1.6 s is:*

*x(1.6) = 5*1.6 = 8.0m*

*y(1.6) = 8.66*1.6-½ 9.8 1.62 = 1.31m*

**Solution e)**

**Solution e)**

*When the coordinate touches the ground and is cancelled, then:*

*y*

**Solution f)**

**Solution f)**

*The horizontal range is the x coordinate just at the instant it hits the ground:*

*x(1.77) = 5*1.77 = 8.85m*

**Example 3**

**Example 3**

*Find the equation of the trajectory with the data from Example 2.*

**Solution **

**Solution**

*The parametric equation of the trajectory is:*

*x*

*y*

*And the Cartesian equation is obtained by isolating t from the first and substituting in the second*

*y = 8.66*(x/5)-½ 9.8 (x/5)^2*

*Simplifying:*

*y = 1.73x – 0.20x^2 *

**References**

**References***P.P. Teodorescu (2007). «Kinematics». Mechanical Systems, Classical Models: Particle Mechanics. Springer. Resnick, Halliday, & Krane (2002). Physics Volume 1. Cecsa, Mexico. Thomas Wallace Wright (1896). Elements of Mechanics Including Kinematics, Kinetics and Statics. E and FN Spon. Wikipedia. Parabolic movement. Retrieved from es.wikipedia.org. Wikipedia. Projectile motion. Retrieved from en.wikipedia.org.*