A body thrown at an angle to the horizon: types of trajectories, formulas

Each of us threw stones into the sky and watched the trajectory of their fall. This is the most common example of the motion of a rigid body in the field of gravitational forces of our planet. In this article, we consider formulas that can be useful for solving problems on the free movement of a body thrown to the horizon at an angle.

The concept of moving to the horizon at an angle

When an initial speed is reported to some solid object, and it starts to climb, and then again falls to the ground, then it is generally accepted that the body moves along a parabolic path. In fact, solving equations for this type of motion shows that the line described by the body in the air is part of the ellipse. However, for practical use, the parabolic approximation is quite convenient and leads to accurate results.

Examples of the movement of a body thrown at an angle to the horizon are a shot of a projectile from a gun’s barrel, a kick on the ball, and even pebbles jumping on the surface of the water (“frogs”), which host international competitions.

The type of motion at an angle is studied by ballistics.

Properties of the type of motion under consideration

When considering the trajectory of the body in the field of gravitational forces of the Earth, the following statements are true:

• knowledge of the initial height, speed and angle to the horizon allows you to calculate the entire trajectory;
• the angle of departure is equal to the angle of incidence of the body, provided that the initial height is zero;
• vertical movement can be considered independently of horizontal movement;

Note that these properties are valid if the friction force during the flight of the body is negligible. In ballistics, when studying the flight of shells, many different factors are taken into account, including friction.

Types of Parabolic Movement

Depending on the height from which the movement begins, at what height it ends, and how the initial speed is directed, the following types of parabolic movement are distinguished:

• Full parabola. In this case, the body is thrown from the surface of the earth, and it falls onto this surface, while describing a complete parabola.
• Half a parabola. Such a schedule of the body’s movement is observed if it is thrown from a certain height h, directing the velocity v parallel to the horizon, that is, at an angle θ = 0 ^o .
• Part of the parabola. Such trajectories arise when the body is thrown at a certain angle θ ≠ 0 ^o , and the difference between the initial and final heights is also not equal to zero (hh ₀ ≠ 0). Most of the trajectories of the movement of objects are of this type. For example, a shot from a cannon standing on a hill, or a basketball player throwing a ball into a basket.

The body motion diagram corresponding to a complete parabola is given above.

Formulas required for calculating

We give formulas to describe the motion of a body thrown at an angle to the horizon. Neglecting the force of friction, and considering only gravity, we can write two equations for the speed of movement of the object:

v _x = v ₀ * cos (θ)

v _y = v ₀ * sin (θ) - g * t

Since gravity is directed vertically downward, it does not change the horizontal component of the velocity v _x , therefore, in the first equality there is no dependence on time. Component v _y, in turn, is affected by gravity, which tells g the acceleration of the body toward the ground (hence the minus sign in the formula).

Now we write the formulas for changing the coordinates of a body thrown at an angle to the horizon:

x = x ₀ + v ₀ * cos (θ) * t

y = y ₀ + v ₀ * sin (θ) * t - g * t 2/2

The initial coordinate x _{0 is} often taken equal to zero. The y ₀ coordinate is nothing more than the height h with which the body is thrown (y ₀ = h).

Now we express time t from the first expression and substitute it into the second, we get:

y = h + tg (θ) * x - g / (2 * v ₀ ² * cos ² (θ)) * x ²

This expression in geometry corresponds to a parabola whose branches are directed downward.

The above equations are sufficient to determine any characteristics of this type of motion. So, their solution leads to the fact that the maximum flight range is achieved if θ = 45 ^o , but the maximum height to which the thrown body rises is reached at θ = 90 ^o .