Body movement vertically downward formula. Free fall of bodies. Movement of a body thrown vertically upward
Questions.
1. Does gravity act on a body thrown up during its ascent?
The force of gravity acts on all bodies, regardless of whether it is thrown up or at rest.
2. With what acceleration does a body thrown up move in the absence of friction? How does the speed of the body change in this case?
3. What does it depend on? highest height lifting a body thrown upward in the case when air resistance can be neglected?
The lifting height depends on initial speed. (For calculations, see previous question).
4. What can be said about the signs of the projections of the vectors of the instantaneous velocity of the body and the acceleration of gravity during the free upward movement of this body?
When a body moves freely upward, the signs of the projections of the velocity and acceleration vectors are opposite.
5. How were the experiments depicted in Figure 30 carried out, and what conclusion follows from them?
For a description of the experiments, see pages 58-59. Conclusion: If only gravity acts on a body, then its weight is zero, i.e. it is in a state of weightlessness.
Exercises.
1. A tennis ball was thrown vertically upward with an initial speed of 9.8 m/s. After what period of time will the speed of the rising ball decrease to zero? How much movement will the ball make from the point of throw?
You know that when any body falls to Earth, its speed increases. For a long time believed that the Earth communicates different bodies various accelerations. Simple observations seem to confirm this.
But only Galileo was able to experimentally prove that in reality this was not the case. Air resistance must be taken into account. It is this that distorts the picture of the free fall of bodies, which could be observed in the absence of the earth’s atmosphere. To test his assumption, Galileo, according to legend, watched the fall from the famous leaning Leaning Tower of Pisa different bodies(cannonball, musket ball, etc.). All these bodies reached the Earth's surface almost simultaneously.
The experiment with the so-called Newton tube is especially simple and convincing. Various objects are placed in a glass tube: pellets, pieces of cork, fluffs, etc. If you now turn the tube so that these objects can fall, then the pellet will flash faster, followed by pieces of cork, and finally the fluff will smoothly fall (Fig. 1, a). But if you pump the air out of the tube, then everything will happen completely differently: the fluff will fall, keeping pace with the pellet and the cork (Fig. 1, b). This means that its movement was delayed by air resistance, which had a lesser effect on the movement of, for example, a traffic jam. When these bodies are only affected by gravity towards the Earth, then they all fall with the same acceleration.
Rice. 1
- Free fall is the movement of a body only under the influence of gravity towards the Earth(without air resistance).
Acceleration imparted to all bodies the globe, called acceleration of free fall. We will denote its module by the letter g. Free fall does not necessarily represent downward movement. If the initial speed is directed upward, then a body in free fall will fly upward for some time, reducing its speed, and only then begin to fall down.
Vertical body movement
- Equation of velocity projection onto the axis 0Y: $\upsilon _(y) =\upsilon _(0y) +g_(y) \cdot t,$
equation of motion along the axis 0Y: $y=y_(0) +\upsilon _(0y) \cdot t+\dfrac(g_(y) \cdot t^(2) )(2) =y_(0) +\dfrac(\upsilon _(y )^(2) -\upsilon _(0y)^(2) )(2g_(y) ) ,$
Where y 0 - initial coordinate of the body; υ y- projection of final speed onto axis 0 Y; υ 0 y- projection of the initial speed onto axis 0 Y; t- time during which the speed changes (s); g y- projection of free fall acceleration onto axis 0 Y.
- If axis 0 Y point upward (Fig. 2), then g y = –g, and the equations will take the form
Rice. 2 Hidden data When the body moves down
- “the body falls” or “the body fell” - υ 0 at = 0.
ground surface, That:
- "the body fell to the ground" - h = 0.
- “the body has reached its maximum height” - υ at = 0.
If we take as the origin of reference ground surface, That:
- "the body fell to the ground" - h = 0;
- “the body was thrown from the ground” - h 0 = 0.
- Rising time body to maximum height t under is equal to the time of falling from this height to the starting point t pad, and the total flight time t = 2t under.
- The maximum lifting height of a body thrown vertically upward from zero height (at maximum height υ y = 0)
Movement of a body thrown horizontally
A special case of the motion of a body thrown at an angle to the horizontal is the motion of a body thrown horizontally. The trajectory is a parabola with its vertex at the throwing point (Fig. 3).
Rice. 3
This movement can be divided into two:
1) uniform movement horizontally with speed υ 0 X (a x = 0)
- velocity projection equation: $\upsilon _(x) =\upsilon _(0x) =\upsilon _(0) $;
- equation of motion: $x=x_(0) +\upsilon _(0x) \cdot t$;
To describe movement along the 0 axis Y formulas for uniformly accelerated vertical motion are applied:
- velocity projection equation: $\upsilon _(y) =\upsilon _(0y) +g_(y) \cdot t$;
- equation of motion: $y=y_(0) +\dfrac(g_(y) \cdot t^(2) )(2) =y_(0) +\dfrac(\upsilon _(y)^(2) )(2g_( y) ) $.
- If axis 0 Y point up then g y = –g, and the equations will take the form:
- Range of flight is determined by the formula: $l=\upsilon _(0) \cdot t_(nad) .$
- Body speed at any time t will be equal (Fig. 4):
where υ X = υ 0 x , υ y = g y t or υ X= υ∙cos α, υ y= υ∙sin α.
Rice. 4
When solving free fall problems
1. Select a reference body, specify the initial and final positions of the body, select the direction of the 0 axes Y and 0 X.
2. Draw a body, indicate the direction of the initial velocity (if it is zero, then the direction of the instantaneous velocity) and the direction of the acceleration of free fall.
3. Write the original equations in projections onto the 0 axis Y(and, if necessary, on axis 0 X)
$\begin(array)(c) (0Y:\; \; \; \; \; \upsilon _(y) =\upsilon _(0y) +g_(y) \cdot t,\; \; \; (1)) \\ () \\ (y=y_(0) +\upsilon _(0y) \cdot t+\dfrac(g_(y) \cdot t^(2) )(2) =y_(0) +\dfrac(\upsilon _(y)^(2) -\upsilon _(0y)^(2) )(2g_(y) ) ,\; \; \; \; (2)) \\ () \ \ (0X:\; \; \; \; \; \upsilon _(x) =\upsilon _(0x) +g_(x) \cdot t,\; \; \; (3)) \\ () \\ (x=x_(0) +\upsilon _(0x) \cdot t+\dfrac(g_(x) \cdot t^(2) )(2).\; \; \; (4)) \end (array)$
4. Find the values of the projections of each quantity
x 0 = …, υ x = …, υ 0 x = …, g x = …, y 0 = …, υ y = …, υ 0 y = …, g y = ….Note. If axis 0 X is directed horizontally, then g x = 0.
5. Substitute the obtained values into equations (1) - (4).
6. Solve the resulting system of equations.
Note. As you develop the skill of solving such problems, point 4 can be done in your head, without writing it in a notebook.
As we already know, the force of gravity acts on all bodies that are on the surface of the Earth and near it. It does not matter whether they are at rest or in motion.
If a body falls freely to the Earth, then it will perform uniformly accelerated motion, and the speed will increase constantly, since the velocity vector and the free fall acceleration vector will be co-directed with each other.
The essence of vertical upward movement
If you throw some body vertically upward, and at the same time, assuming that there is no air resistance, then we can assume that it also performs uniformly accelerated motion, with the acceleration of free fall, which is caused by gravity. Only in this case, the speed that we gave to the body during the throw will be directed upward, and the acceleration of free fall will be directed downward, that is, they will be oppositely directed to each other. Therefore, the speed will gradually decrease.
After some time, a moment will come when the speed becomes zero. At this moment the body will reach its maximum height and stop for a moment. Obviously, the greater the initial speed we give to the body, the greater the height it will rise by the time it stops.
- Next, the body will begin to fall down uniformly under the influence of gravity.
How to solve problems
When you are faced with tasks on the upward movement of a body, in which air resistance and other forces are not taken into account, but it is believed that only the force of gravity acts on the body, then since the movement is uniformly accelerated, you can apply the same formulas as for rectilinear uniformly accelerated moving with some initial speed V0.
Since in this case the acceleration ax is the acceleration of free fall of the body, then ax is replaced by gx.
- Vx=V0x+gx*t,
- Sx=V(0x)*t+(gx*t^2)/2.
It is also necessary to take into account that when moving upward, the free fall acceleration vector is directed downward, and the velocity vector is directed upward, that is, they are in different directions, and therefore, their projections will have different signs.
For example, if the Ox axis is directed upward, then the projection of the velocity vector when moving upward will be positive, and the projection of the free fall acceleration will be negative. This must be taken into account when substituting values into formulas, otherwise you will get a completely incorrect result.
This video tutorial is intended for self-study topic “Motion of a body thrown vertically upward.” In this lesson, students will gain an understanding of the motion of a body in free fall. The teacher will talk about the movement of a body thrown vertically upward.
In the previous lesson, we looked at the issue of the movement of a body that was in free fall. Let us recall that free fall (Fig. 1) is a movement that occurs under the influence of gravity. The force of gravity is directed vertically downwards along the radius towards the center of the Earth, acceleration of gravity at the same time equal to .
Rice. 1. Free fall
How will the movement of a body thrown vertically up differ? It will differ in that the initial speed will be directed vertically upward, i.e., it can also be counted along the radius, but not towards the center of the Earth, but, on the contrary, from the center of the Earth upward (Fig. 2). But the acceleration of free fall, as you know, is directed vertically downward. This means that we can say the following: the upward movement of a body in the first part of the path will be a slow motion, and this slow motion will also occur with the acceleration of free fall and also under the influence of gravity.
Rice. 2 Movement of a body thrown vertically upward
Let's look at the picture and see how the vectors are directed and how this fits into the reference frame.
Rice. 3. Movement of a body thrown vertically upward
In this case, the reference frame is connected to the ground. Axis Oy is directed vertically upward, just like the initial velocity vector. The body is acted upon by a force of gravity directed downward, which imparts to the body the acceleration of free fall, which will also be directed downward.
The following thing can be noted: the body will move slowly, will rise to a certain height, and then will start quickly fall down.
We have indicated the maximum height.
The motion of a body thrown vertically upward occurs near the Earth's surface, when the acceleration of free fall can be considered constant (Fig. 4).
Rice. 4. Near the Earth's surface
Let us turn to the equations that make it possible to determine the speed, instantaneous speed and distance traveled during the movement in question. The first equation is the velocity equation: . The second equation is the equation of motion for uniformly accelerated motion: .
Rice. 5. Axis Oy upward
Let's consider the first frame of reference - the frame of reference associated with the Earth, the axis Oy directed vertically upward (Fig. 5). The initial speed is also directed vertically upward. In the previous lesson, we already said that the acceleration of gravity is directed downwards along the radius towards the center of the Earth. So, if we now bring the velocity equation to this reference frame, we get the following: .
This is a projection of speed at a certain point in time. The equation of motion in this case has the form: 
.
Rice. 6. Axle Oy pointing down
Let's consider another frame of reference, when the axis Oy directed vertically downwards (Fig. 6). What will change from this?
. The projection of the initial velocity will have a minus sign, since its vector is directed upward, and the axis of the selected reference system is directed downward. In this case, the acceleration of gravity will have a plus sign, because it is directed downward. Equation of motion: 
.
Another very important concept to consider is the concept of weightlessness.
Definition.Weightlessness- a state in which a body moves only under the influence of gravity.
Definition. Weight- the force with which a body acts on a support or suspension due to attraction to the Earth.
Rice. 7 Illustration for determining weight
If a body near the Earth or at a short distance from the Earth’s surface moves only under the influence of gravity, then it will not affect the support or suspension. This state is called weightlessness. Very often, weightlessness is confused with the concept of the absence of gravity. In this case, it is necessary to remember that weight is the action on the support, and weightlessness- this is when there is no effect on the support. Gravity is a force that always acts near the surface of the Earth. This force is the result of gravitational interaction with the Earth.
Let's pay attention to one more important point, associated with the free fall of bodies and movement vertically upward. When a body moves upward and moves with acceleration (Fig. 8), an action occurs that leads to the fact that the force with which the body acts on the support exceeds the force of gravity. When this happens, the state of the body is called overload, or the body itself is said to be under overload.
Rice. 8. Overload
Conclusion
The state of weightlessness, the state of overload are extreme cases. Basically, when a body moves on a horizontal surface, the weight of the body and the force of gravity most often remain equal to each other.
Bibliography
- Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. school - M.: Education, 1992. - 191 p.
- Sivukhin D.V. General physics course. - M.: State Publishing House of Technology
- theoretical literature, 2005. - T. 1. Mechanics. - P. 372.
- Sokolovich Yu.A., Bogdanova G.S. Physics: A reference book with examples of problem solving. - 2nd edition, revision. - X.: Vesta: Ranok Publishing House, 2005. - 464 p.
- Internet portal “eduspb.com” ()
- Internet portal “physbook.ru” ()
- Internet portal “phscs.ru” ()
Homework
The body itself, as is known, does not move upward. It needs to be “thrown”, that is, it needs to be given a certain initial speed directed vertically upward.
A body thrown upward moves, as experience shows, with the same acceleration as a freely falling body. This acceleration is equal and directed vertically downward. The motion of a body thrown upward is also rectilinear uniformly accelerated motion, and the formulas that were written for the free fall of a body are also suitable for describing the motion of a body thrown upward. But when writing formulas, it is necessary to take into account that the acceleration vector is directed against the initial velocity vector: the speed of the body in absolute value does not increase, but decreases. Therefore, if the coordinate axis is directed upward, the projection of the initial velocity will be positive, and the projection of acceleration will be negative, and the formulas will take the form:
Since a body thrown upward moves with decreasing speed, a moment will come when the speed becomes zero. At this moment the body will be at its maximum height. Substituting the value into formula (1) we get:
From here you can find the time it takes for the body to rise to its maximum height:
The maximum height is determined from formula (2).
Substituting into the formula we get
After the body reaches a height it will begin to fall down; the projection of its speed will become negative, and according to absolute value will increase (see formula 1), while the height will decrease over time according to formula (2) at
Using formulas (1) and (2), it is easy to verify that the speed of the body at the moment of its fall to the ground or generally to where it was thrown from (at h = 0) is equal in absolute value to the initial speed and the time of fall of the body is equal to the time of its rise.
The fall of a body can be considered separately as free fall bodies from a height Then we can use the formulas given in the previous paragraph.
Task. A body is thrown vertically upward at a speed of 25 m/sec. What is the speed of the body after 4 seconds? What displacement will the body make and what is the length of the path traveled by the body during this time? Solution. The speed of the body is calculated by the formula
By the end of the fourth second
The sign means that the speed is directed against the coordinate axis directed upward, i.e. at the end of the fourth second the body was already moving downwards, having passed through highest point of its rise.
We find the amount of movement of the body using the formula
This movement is counted from the place from which the body was thrown. But at that moment the body was already moving down. Therefore, the length of the path traveled by the body is equal to the maximum height of the rise plus the distance by which it managed to fall down:
We calculate the value using the formula
Substituting the values we get: sec
Exercise 13
1. An arrow is shot vertically upward from a bow at a speed of 30 m/sec. How high will it rise?
2. A body thrown vertically upward from the ground fell after 8 seconds. Find to what height it rose and what was its initial speed?
3. From a spring gun located at a height of 2 m above the ground, a ball flies vertically upward at a speed of 5 m/sec. Determine which maximum height it will rise and what speed the ball will have at the moment it falls to the ground. How long was the ball in flight? What is its displacement during the first 0.2 seconds of flight?
4. A body is thrown vertically upward at a speed of 40 m/sec. At what height will it be after 3 and 5 seconds and what speeds will it have? Accept
5 Two bodies are thrown vertically upward with different initial velocities. One of them reached four times greater height than another. How many times was its initial speed greater than the initial speed of the other body?
6. A body thrown upward flies past the window at a speed of 12 m/sec. At what speed will it fly down past the same window?