TRAJECTORY IN PHYSICS: CHARACTERISTICS, TYPES, EXAMPLES AND EXERCISES - PHYSICAL - 2025

The trajectory in physics is the curve that a mobile describes as it passes through successive points during its movement. Since it can take many variants, so will the paths that the mobile can follow.

To get from one place to another, a person can take different paths and different ways: on foot through the sidewalks in streets and avenues, or arriving by car or motorcycle on a highway. During a walk through the forest, the hiker can follow a complicated path that includes turns, going up or down in level and even passing through the same point several times.

Figure 1. Uniting the end points of each position vector the path followed by the particle is obtained. Source: Algarabia

If the points through which the mobile is traveling follow a straight line, the trajectory will be rectilinear. This is the simplest path, as it is one-dimensional. Specifying the position requires a single coordinate.

But the mobile can follow a curvilinear path, being able to be closed or open. In these cases, tracking the position requires two or three coordinates. These are movements in the plane and in space respectively. This has to do with links: limiting material conditions of movement. Some examples are:

- The orbits that describe the planets around the sun are closed paths in the shape of an ellipse. Although, in some cases, they can be approximated to a circular, as in the case of Earth.

- The ball that the goalkeeper kicks in a goal kick follows a parabolic trajectory.

- A bird in flight describes curvilinear trajectories in space, because in addition to moving on a plane, it can go up or down in level at will.

The trajectory in physics can be expressed mathematically when the position of the mobile is known at any instant of time. Let r be the position vector, which in turn has x, y and z coordinates in the most general case of a three-dimensional motion. Knowing the function r (t) the trajectory will be completely determined.

Types

In general terms, the trajectory can be a rather complicated curve, especially if you want to express it mathematically. For this reason, it begins with the simplest models, where the mobiles travel on a straight line or on a plane, which can be the floor or any other suitable one:

Movements in one, two and three dimensions

The most studied trajectories are:

- Rectilinear, when traveling on a straight horizontal, vertical or inclined line. A ball thrown vertically upward follows this path, or an object sliding down an incline follows. They are one-dimensional movements, a single coordinate being enough to determine their position completely.

- Parabolic, in which the mobile describes a parabola arc. It is frequent, since any object thrown obliquely under the action of gravity (a projectile) follows this trajectory. To specify the position of the mobile you have to give two coordinates: x and y.

- Circular, occurs when the moving particle follows a circle. It is also common in nature and in daily practice. Many everyday objects follow a circular path such as tires, machinery parts, and orbiting satellites, to give a few examples.

- Elliptical, the object moves following an ellipse. As said at the beginning, it is the path followed by the planets in orbit around the sun.

- Hyperbolic, astronomical objects under the action of a central force (gravity), can follow elliptical (closed) or hyperbolic (open) trajectories, these being less frequent than the former.

- Helical, or spiral movement, like that of a bird ascending in a thermal current.

- Sway or pendulum, the mobile describes an arc in back and forth movements.

Examples

The trajectories described in the previous section are very useful to quickly get an idea of ​​how an object is moving. In any case, it is necessary to clarify that the trajectory of a mobile depends on the location of the observer. This means that the same event can be seen in different ways, depending on where each person is.

For example, a girl pedals at a constant speed and throws a ball upwards. She observes that the ball describes a rectilinear path.

However, for an observer standing on the road who sees it pass, the ball will have a parabolic movement. For him, the ball was initially thrown with an inclined speed, a result of the speed upward by the girl's hand plus the speed of the bicycle.

Figure 2. This animation shows the vertical throw of a ball made by a girl riding a bicycle, as she sees it (rectilinear trajectory) and as an observer sees it (parabolic trajectory). (Prepared by F. Zapata).

Path of a mobile in explicit, implicit and parametric way

- Explicit, directly specifying the curve or locus given by the equation y (x)

- Implicit, in which a curve is expressed as f (x, y, z) = 0

- Parametric, in this way the coordinates x, y and z are given as a function of a parameter that, in general, is chosen as time t. In this case, the trajectory is composed of the functions: x (t), y (t) and z (t).

Next, two trajectories that have been widely studied in kinematics are detailed: the parabolic trajectory and the circular trajectory.

Tilted launch into the void

An object (the projectile) is thrown at an angle a with the horizontal and with initial velocity v _o as shown in the figure. Air resistance is not taken into account. The movement can be treated as two independent and simultaneous movements: one horizontal with constant speed and the other vertical under the action of gravity.

These equations are the parametric equations of projectile launch. As explained above, they have a common parameter t, which is time.

The following can be seen in the right triangle in the figure:

Figure 3. Parabolic trajectory followed by a projectile, in which the components of the velocity vector are shown. H is the maximum height and R is the maximum horizontal reach. Source: Ayush12gupta

Substituting these equations containing the launch angle into the parametric equations results:

Equation of the parabolic path

The explicit equation of the path is found by solving t from the equation for x (t) and substituting in the equation for y (t). To facilitate algebraic work, it can be assumed that the origin (0,0) is located at the launch point and thus x _o = y _o = 0.

This is the equation of the path in explicit form.

Circular path

A circular path is given by:

Figure 4. A particle moves in a circular path on the plane. Source: modified by F. Zapata from Wikimedia Commons.

Here x _or yy _o represent the center of the circumference described by the mobile and R is its radius. P (x, y) is a point on the path. From the shaded right triangle (figure 3) it can be seen that:

The parameter, in this case, is the swept angle θ, called the angular displacement. In the particular case that the angular velocity ω (angle swept per unit time) is constant, it can be stated that:

Where θ _o is the initial angular position of the particle, which if taken as 0, reduces to:

In such case, the time returns to the parametric equations as:

The unit vectors i and j are very convenient for writing the position function of an object r (t). They indicate the directions on the x-axis and on the y-axis respectively. In its terms, the position of a particle that describes a Uniform Circular Motion is:

r (t) = R.cos ω t i + R. sin ω t j

Solved exercises

Resolved exercise 1

A cannon can fire a bullet with a velocity of 200 m / s and an angle of 40º with respect to the horizontal. If the throw is on flat ground and air resistance is neglected, find:

a) The equation of the path y (x)..

b) The parametric equations x (t) and y (t).

c) The horizontal range and the time the projectile lasts in the air.

d) The height at which the projectile is when x = 12,000 m

Solution to)

a) To find the trajectory, the values ​​given in the equation y (x) of the previous section are substituted:

Solution b)

b) The launch point is chosen at the origin of the coordinate system (0,0):

Solution c)

c) To find the time that the projectile lasts in the air, let y (t) = 0, where the launch is made on flat ground:

The maximum horizontal reach is found by substituting this value in x (t):

Another way to find x _max directly is by setting y = 0 in the equation of the path:

There is a small difference due to rounding of the decimals.

Solution d)

d) To find the height when x = 12000 m, this value is substituted directly in the equation of the path:

Exercise solved 2

The position function of an object is given by:

r (t) = 3t i + (4 -5t ²) j m

Find:

a) The equation for the path. What curve is it?

b) The initial position and the position when t = 2 s.

c) The displacement made after t = 2 s.

Solution

a) The position function has been given in terms of the unit vectors i and j, which respectively determine the direction in the x and y axes, therefore:

The equation of the path y (x) is found by solving t from x (t) and substituting in y (t):

b) The initial position is: r (2) = 4 j m; the position at t = 2 s is r (2) = 6 i -16 j m

c) The displacement D r is the subtraction of the two position vectors:

Exercise solved 3

The Earth has a radius R = 6300 km and it is known that the period of rotation of its movement around its axis is one day. Find:

a) The equation of the trajectory of a point on the earth's surface and its position function.

b) The speed and acceleration of that point.

Solution to)

a) The position function for any point in circular orbit is:

r (t) = R.cos ω t i + R. sin ω t j

We have the radius of the Earth R, but not the angular velocity ω, however it can be calculated from the period, knowing that for circular motion it is valid to say that:

The period of the movement is: 1 day = 24 hours = 1440 minutes = 86 400 seconds, therefore:

Substituting in the position function:

r (t) = R.cos ω t i + R. sin ω t j = 6300 (cos 0.000023148t i + sin 0.000023148t j) Km

The path in parametric form is:

Solution b)

b) For circular motion, the magnitude of the linear velocity v of a point is related to the angular velocity w by:

Even being a motion with constant speed of 145.8 m / s, there is an acceleration that points towards the center of the circular orbit, in charge of keeping the point in rotation. It is the centripetal acceleration at _c, given by:

References

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