Determining the initial velocity of a body thrown horizontally. Topic: Studying the motion of a body thrown horizontally
Subject: Study of the motion of a body thrown horizontally.
Goal of the work: to investigate the dependence of the flight range of a body thrown horizontally on the height from which it began to move.
Equipment:
- tripod with clutch;
- steel ball;
- copy paper;
- guide rail;
- ruler;
- scotch.
If a body is thrown horizontally from a certain height, then its motion can be considered as a motion of inertia along the horizontal and uniformly accelerated motion along the vertical.
Horizontally, the body moves by inertia in accordance with Newton's first law, since, apart from the resistance force from the side of the air, which is not taken into account, no other forces act on it in this direction. The force of air resistance can be neglected because a short time the flight of a body thrown from a small height, the action of this force will not have a noticeable effect on the movement.
The force of gravity acts on the body vertically, which imparts acceleration to it. g(acceleration of gravity).
Considering the movement of the body in such conditions as the result of two independent movements horizontally and vertically, it is possible to establish the dependence of the flight range of the body on the height from which it is thrown. Considering that the speed of the body V at the time of the throw is directed horizontally, and there is no vertical component of the initial velocity, then the fall time can be found using the basic equation of uniformly accelerated motion:
Where .
During the same time, the body will have time to fly horizontally, moving uniformly, the distance S=Vt. Substituting the already found flight time into this formula, we obtain the desired dependence of the flight range on altitude and speed:
From the resulting formula, it can be seen that the throw distance is proportional to the square root of the height from which the throw is made. For example, if the altitude is quadrupled, the flight range will double; with a ninefold increase in height, the range will increase by a factor of three, and so on.
This conclusion can be confirmed more strictly. Let when thrown from a height H1 range will be S1, when thrown at the same speed from a height H 2 \u003d 4H 1 range will be S2
According to the formula
: 
And 
Dividing the second equation by the first:
or S2 = 2S1
This dependence, obtained theoretically from the equations of uniform and uniformly accelerated motion, is verified experimentally in the work.
The paper investigates the movement of the ball, which rolls down from the stop from the chute of the inverted guide rail. The guide rail is mounted on a tripod, the design allows you to give the ball a horizontal direction of speed at a certain height above the table. This ensures the horizontal direction of the speed of the ball at the moment of the beginning of its free flight.
Two series of experiments are carried out, in which the heights of the separation of the ball differ by a factor of four, and the distances are measured S1 And S2, on which the ball is removed from the guide rail horizontally to the point of contact with the table. To reduce the influence on the result of side factors, the average value of the distances is determined S 1av And S 2av. Comparing the average distances obtained in each series of experiments, they conclude how true the FORMULA equality is.
Instructions for work
1. Fix the guide rail upside down on the tripod shaft so that the sleeve prevents it from falling down from the tripod. Place the point of separation of the ball from the same guide rail at a height of about 9 cm from the surface of the table. Place carbon paper where the ball is supposed to fall on the table.
2. Prepare a table to record the results of measurements and calculations.
| experience number | H 1 cm | S1 , cm | S 1av , cm | H 2 , cm | S2 , cm | S 2cr , cm |
| 1 |
3. Test run the ball from the start of the guide rail groove. Determine where the ball falls on the table. The ball should fall into the middle part of the film. Adjust the position of the film if necessary. Stick the film to the table with a piece of tape.
4. Using a ruler, measure the height of the ball's breakaway point above the table H1. Using a ruler set vertically, mark on the surface of the table a point (for example, with a piece of adhesive tape), above which the point of separation of the ball from the guide rail is located.
5. Run the ball from the start of the guide rail groove and measure the distance on the table surface S1 from the point of separation of the ball from the guide rail, to the mark left on the film by the ball when it falls.
6. Repeat the ball launch 5-6 times. In order for the speed with which the ball flies off the guide rail to be the same in all experiments, it is launched from the same point from the beginning of the groove of the guide rail.
7. Calculate the average value of the distance S 1av.
8. Increase the ball lift off the guide rail by four times. Make sure the condition is met: H 2 \u003d 4H 1.
9. Repeat a series of ball launches from the start of the guide rail groove. For each start, measure the distance S2 and calculate the mean S 2cr.
10. Check if equality is true S 2cr = 2S 1av . Specify possible cause discrepancies in results.
11. Make a conclusion about the dependence of the flight range of a horizontally thrown body on the height of the throw, from which the body began to move.
Laboratory work (experimental task)
DETERMINATION OF THE INITIAL SPEED OF THE BODY,
THROWN HORIZONTALLY
Equipment: pencil eraser (eraser), measuring tape, wooden blocks.
Goal of the work: experimentally determine the value of the initial velocity of a body thrown horizontally. Assess the credibility of the result.
Equations of motion of a material point in projections onto the horizontal axis 0 X and vertical axis 0 y look like this:
The horizontal component of the velocity during the movement of a body thrown horizontally does not change, therefore, the path of the body during the free flight of the body horizontally is determined as follows: https://pandia.ru/text/79/468/images/image004_28.gif" width="112 " height="44 src="> From this equation, find the time and substitute the resulting expression in the previous formula. Now you can get the calculation formula for finding the initial speed of a body thrown horizontally:
Work order
1. Prepare sheets for the report on the work done with preliminary entries.
2. Measure the table height.
3. Place the eraser on the edge of the table. Click to move it in a horizontal direction.
4. Mark the spot where the elastic will reach the floor. Measure the distance from the point on the floor where the edge of the table is projected to the point where the elastic band falls on the floor.
5. Change the flight height of the eraser by placing a wooden block (or box) under it on the edge of the table. Do the same for the new case.
6. Conduct at least 10 experiments, enter the measurement results in the table, calculate the initial speed of the eraser, assuming the free fall acceleration is 9.81 m/s2.
Table of measurement and calculation results
|
experience | Body flight height | body flight range | Initial body speed | Absolute speed error |
h | s | v 0 | D v 0 |
|
Average |
7. Calculate the magnitude of the absolute and relative errors of the initial velocity of the body, draw conclusions about the work done.
Control questions
1. A stone is thrown vertically upwards and the first half of the way moves uniformly slow, and the second half - uniformly accelerated. Does this mean that its acceleration is negative on the first half of the path, and positive on the second?
2. How does the velocity modulus of a body thrown horizontally change?
3. In which case the object that fell out of the car window will fall to the ground earlier: when the car is standing still or when it is moving: Neglect air resistance.
4. In what case is the module of the displacement vector of a material point the same as the path?
Literature:
1.Giancoli D. Physics: In 2 vols. T. 1: Per. from English - M.: Mir, 1989, p. 89, task 17.
2. , Experimental tasks in physics. Grades 9-11: a textbook for students of educational institutions. - M .: Verbum-M, 2001, p. 89.
Here is the initial speed of the body, is the speed of the body at the moment of time t, s- horizontal flight distance, h is the height above the ground from which a body is thrown horizontally with a speed .
1.1.33. Kinematic equations of velocity projection:
1.1.34. Kinematic coordinate equations:
1.1.35. body speed at the time t:
In the moment falling to the ground y=h, x = s(Fig. 1.9).
1.1.36. Maximum horizontal flight range:
1.1.37. Height above ground from which the body is thrown
horizontally:
Motion of a body thrown at an angle α to the horizon
with initial speed
1.1.38. The trajectory is a parabola(Fig. 1.10). Curvilinear movement along a parabola is due to the result of adding two rectilinear movements: uniform movement along the horizontal axis and equally variable movement along the vertical axis.
 |
| Rice. 1.10 |
( is the initial speed of the body, are the projections of the velocity on the coordinate axes at the moment of time t, is the flight time of the body, hmax- the maximum height of the body, smax is the maximum horizontal flight distance of the body).
1.1.39. Kinematic projection equations:
; 
1.1.40. Kinematic coordinate equations:
; 
1.1.41. The height of the body lift to the top point of the trajectory:
At the moment of time , (Figure 1.11).
1.1.42. Maximum body height: 
1.1.43. Body flight time: 
At the point in time , (Fig. 1.11).
1.1.44. Maximum horizontal flight range of the body:
1.2. Basic equations of classical dynamics
Dynamics(from Greek. dynamic- force) - a branch of mechanics devoted to the study of the movement of material bodies under the action of forces applied to them. Classical dynamics are based on Newton's laws . All equations and theorems necessary for solving problems of dynamics are obtained from them.
1.2.1. Inertial Reporting System - it is a frame of reference in which the body is at rest or moving uniformly and in a straight line.
1.2.2. Force is the result of the interaction of the body with environment. One of the simplest definitions of force: the influence of a single body (or field) that causes acceleration. Currently, four types of forces or interactions are distinguished:
· gravitational(manifested in the form of forces gravity);
· electromagnetic(existence of atoms, molecules and macrobodies);
· strong(responsible for the connection of particles in nuclei);
· weak(responsible for the decay of particles).
1.2.3. The principle of superposition of forces: if several forces act on a material point, then the resulting force can be found by the rule of vector addition:
.
The mass of a body is a measure of the inertia of a body. Any body resists when trying to set it in motion or change the module or direction of its speed. This property is called inertia.
1.2.5. Pulse(momentum) is the product of the mass T body by its speed v:
1.2.6. Newton's first law: Any material point (body) maintains a state of rest or uniform rectilinear motion until the impact from other bodies causes her (him) to change this state.
1.2.7. Newton's second law(basic equation of the dynamics of a material point): the rate of change of the momentum of the body is equal to the force acting on it (Fig. 1.11):
 |  |
| Rice. 1.11 | Rice. 1.12 |
The same equation in projections onto the tangent and normal to the point trajectory:
And 
.
1.2.8. Newton's third law: the forces with which two bodies act on each other are equal in magnitude and opposite in direction (Fig. 1.12):
1.2.9. Law of conservation of momentum for a closed system: the momentum of a closed system does not change in time (Fig. 1.13):
,
Where P is the number of material points (or bodies) included in the system.
 |  |
| Rice. 1.13 |
The law of conservation of momentum is not a consequence of Newton's laws, but is fundamental law of nature, which knows no exceptions, and is a consequence of the homogeneity of space.
1.2.10. The basic equation of the dynamics of the translational motion of a system of bodies:
where is the acceleration of the center of inertia of the system; is the total mass of the system from P material points.
1.2.11. Center of mass of the system material points (Fig. 1.14, 1.15):
.
The law of motion of the center of mass: the center of mass of the system moves like a material point, the mass of which is equal to the mass of the entire system and which is affected by a force equal to the vector sum of all forces acting on the system.
1.2.12. Impulse of the body system:
where is the speed of the center of inertia of the system.
 |  |
| Rice. 1.14 | Rice. 1.15 |
1.2.13. Theorem on the motion of the center of mass: if the system is in an external stationary uniform force field, then no actions inside the system can change the motion of the center of mass of the system:
.
1.3. Forces in mechanics
1.3.1. Body weight relationship with gravity and support reaction:
Free fall acceleration (Fig. 1.16).
 |
| Rice. 1.16 |
Weightlessness is a state in which the weight of a body is zero. In a gravitational field, weightlessness occurs when a body moves only under the action of gravity. If a = g, That p=0.
1.3.2. Relationship between weight, gravity and acceleration:
1.3.3. sliding friction force(Fig. 1.17):
where is the coefficient of sliding friction; N is the force of normal pressure.
1.3.5. Basic ratios for a body on an inclined plane(Fig. 1.19). :
· friction force: 
;
· resultant force: ;
· rolling force: 
;
· acceleration:
 |
| Rice. 1.19 |
1.3.6. Hooke's law for a spring: spring extension X proportional to the force of elasticity or external force:
Where k- spring stiffness.
1.3.7. Potential energy of an elastic spring:
1.3.8. The work done by the spring:
1.3.9. Voltage- measure internal forces arising in a deformable body under the influence of external influences(Fig. 1.20):
where is the cross-sectional area of the rod, d is its diameter, is the initial length of the rod, is the increment of the rod length.
 |  |
| Rice. 1.20 | Rice. 1.21 |
1.3.10. Strain diagram - plot of normal stress σ = F/S on relative elongation ε = Δ l/l when stretching the body (Fig. 1.21).
1.3.11. Young's modulus is the value characterizing the elastic properties of the rod material:
1.3.12. Bar length increment proportional to voltage:
1.3.13. Relative longitudinal tension (compression):
1.3.14. Relative transverse tension (compression):
where is the initial transverse dimension of the rod.
1.3.15. Poisson's ratio- the ratio of the relative transverse tension of the rod to the relative longitudinal tension:
1.3.16. Hooke's law for a rod: relative increment of the length of the rod is directly proportional to the stress and inversely proportional to the Young's modulus:
1.3.17. Bulk potential energy density:
1.3.18. Relative shift ( pic1.22, 1.23 ):
where is the absolute shift.
 |  |
| Rice. 1.22 | Fig.1.23 |
1.3.19. Shear modulusG- a value that depends on the properties of the material and is equal to such a tangential stress at which (if such huge elastic forces were possible).
1.3.20. Tangential elastic stress:
1.3.21. Hooke's law for shear:
1.3.22. Specific potential energy bodies in shear:
1.4. Non-inertial frames of reference
Non-inertial frame of reference is an arbitrary frame of reference that is not inertial. Examples of non-inertial systems: a system moving in a straight line with constant acceleration, as well as a rotating system.
The forces of inertia are due not to the interaction of bodies, but to the properties of the non-inertial frames of reference themselves. Newton's laws do not apply to inertial forces. The forces of inertia are not invariant with respect to the transition from one frame of reference to another.
In a non-inertial system, you can also use Newton's laws if you introduce inertial forces. They are fictitious. They are introduced specifically to use Newton's equations.
1.4.1. Newton's equation for non-inertial frame of reference
where is the acceleration of a body of mass T relative to the non-inertial system; – force of inertia is a fictitious force due to the properties of the frame of reference.
1.4.2. Centripetal force- inertia force of the second kind, applied to a rotating body and directed along the radius to the center of rotation (Fig. 1.24):
,
where is the centripetal acceleration.
1.4.3. Centrifugal force- the force of inertia of the first kind, applied to the connection and directed along the radius from the center of rotation (Fig. 1.24, 1.25):
,
where is the centrifugal acceleration.
  |  |
| Rice. 1.24 | Rice. 1.25 |
1.4.4. Gravity acceleration dependence g from the latitude of the area is shown in fig. 1.25.
Gravity is the result of the addition of two forces: and; Thus, g(and hence mg) depends on latitude:
,
where ω is the angular velocity of the Earth's rotation.
1.4.5. Coriolis force- one of the forces of inertia that exists in a non-inertial frame of reference due to rotation and the laws of inertia, which manifests itself when moving in a direction at an angle to the axis of rotation (Fig. 1.26, 1.27).
where is the angular velocity of rotation.
 |  |
| Rice. 1.26 | Rice. 1.27 |
1.4.6. Newton's equation for non-inertial frames of reference, taking into account all forces, takes the form
where is the force of inertia due to the translational motion of a non-inertial frame of reference; And 
– two inertial forces due to the rotational motion of the reference frame; is the acceleration of the body relative to the non-inertial frame of reference.
1.5. Energy. Job. Power.
Conservation laws
1.5.1. Energy- universal measure various forms motion and interaction of all kinds of matter.
1.5.2. Kinetic energy is the function of the state of the system, determined only by the speed of its movement:
The kinetic energy of a body is a scalar physical quantity equal to half the product of the mass m body per square of its speed.
1.5.3. Theorem on the change in kinetic energy. The work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body, or, in other words, the change in the kinetic energy of the body is equal to the work A of all forces acting on the body.
1.5.4. Relationship between kinetic energy and momentum:
1.5.5. Force work is a quantitative characteristic of the process of energy exchange between interacting bodies. Work in mechanics 
.
1.5.6. Work of a constant force:
If a body is moving in a straight line and a constant force is acting on it F, which makes a certain angle α with the direction of movement (Fig. 1.28), then the work of this force is determined by the formula:
,
Where F is the modulus of force, ∆r is the modulus of displacement of the force application point, is the angle between the direction of force and displacement.
If< /2, то работа силы положительна. Если >/2, then the work done by the force is negative. At = /2 (the force is directed perpendicular to the displacement), then the work of the force is zero.
| Rice. 1.28 | Rice. 1.29 |
Work of constant force F when moving along the axis x at a distance (Fig. 1.29) is equal to the force projection on this axis multiplied by displacement:
.
On fig. 1.27 shows the case when A < 0, т.к. >/2 - obtuse angle.
1.5.7. elementary work d A strength F on elementary displacement d r is called a scalar physical quantity equal to the scalar product of force and displacement:
1.5.8. Variable force work on the trajectory section 1 - 2 (Fig. 1.30):
  |
| Rice. 1.30 |
1.5.9. Instant Power is equal to the work done per unit of time:
.
1.5.10. Average power for a period of time:
1.5.11. Potential energy body at a given point is a scalar physical quantity, equal to the work done by the potential force when moving the body from this point to another taken as the zero of the potential energy reference.
Potential energy is determined up to some arbitrary constant. This is not reflected in the physical laws, since they include either the difference in potential energies in two positions of the body or the derivative of the potential energy with respect to coordinates.
Therefore, the potential energy in a certain position is considered equal to zero, and the energy of the body is measured relative to this position (zero reference level).
1.5.12. The principle of minimum potential energy. Any closed system tends to move to a state in which its potential energy is minimal.
1.5.13. The work of conservative forces is equal to the change in potential energy
.
1.5.14. Vector circulation theorem: if the circulation of any force vector is zero, then this force is conservative.
The work of conservative forces along a closed loop L is zero(Fig. 1.31):
 |  |
| Rice. 1.31 |
1.5.15. Potential energy of gravitational interaction between the masses m And M(Fig. 1.32):
1.5.16. Potential energy of a compressed spring(Fig. 1.33):
  |   |
| Rice. 1.32 | Rice. 1.33 |
1.5.17. Total mechanical energy of the system is equal to the sum of kinetic and potential energies:
E = E to + E P.
1.5.18. Potential energy of the body on high h above the ground
E n = mgh.
1.5.19. Relationship between potential energy and force:
Or 
or
1.5.20. Law of conservation of mechanical energy(for a closed system): the total mechanical energy of a conservative system of material points remains constant:
1.5.21. Law of conservation of momentum for a closed system of bodies:
1.5.22. Law of conservation of mechanical energy and momentum with absolutely elastic central impact (Fig. 1.34):
Where m 1 and m 2 - masses of bodies; and are the speeds of the bodies before the impact.
 |  |
| Rice. 1.34 | Rice. 1.35 |
1.5.23. Body speeds after a perfectly elastic impact (Fig. 1.35):
.
1.5.24. Body speed after a completely inelastic central impact (Fig. 1.36):
1.5.25. Law of conservation of momentum when the rocket is moving (Fig. 1.37):
where and are the mass and speed of the rocket; and the mass and velocity of the ejected gases.
 |  |
|
| Rice. 1.36 | Rice. 1.37 | |
1.5.26. Meshchersky equation for the rocket.
Grade 10
Lab #1
Definition of free fall acceleration.
Equipment: a ball on a thread, a tripod with a clutch and a ring, a measuring tape, a clock.
Work order
The model of a mathematical pendulum is a small-radius metal ball suspended on a long thread.
pendulum length determined by the distance from the suspension point to the center of the ball (according to formula 1)
Where - the length of the thread from the point of suspension to the place where the ball is attached to the thread; is the diameter of the ball. Thread length measured with a ruler, ball diameter - caliper.
Leaving the thread taut, the ball is removed from the equilibrium position by a distance that is very small compared to the length of the thread. Then the ball is released without giving it a push, and at the same time the stopwatch is turned on. Determine the period of timet , during which the pendulum makesn = 50 complete oscillations. The experiment is repeated with two other pendulums. The obtained experimental results ( ) are entered in the table.
Measurement number
t , With
T, s
g, m/s
By formula (2)
calculate the period of oscillation of the pendulum, and from the formula
(3) calculate the acceleration of a freely falling bodyg .
(3)
The measurement results are entered in the table.
Calculate the arithmetic mean from the measurement results and mean absolute error .The final result of measurements and calculations is expressed as 
.
Grade 10
Lab work № 2
Studying the motion of a body thrown horizontally
Goal of the work: measure the initial speed of a body thrown horizontally, to investigate the dependence of the flight range of a body thrown horizontally on the height from which it began to move.
Equipment: tripod with sleeve and clamp, curved chute, metal ball, a sheet of paper, a sheet of carbon paper, a plumb line, a measuring tape.
Work order
The ball rolls down a curved chute, the lower part of which is horizontal. Distanceh from the bottom edge of the chute to the table should be 40 cm. The jaws of the clamp should be located near the top end of the chute. Place a sheet of paper under the chute, pressing it down with a book so that it does not move during the experiments. Mark a point on this sheet with a plumb line.A located on the same vertical with the lower end of the gutter. Release the ball without pushing. Note (approximately) the spot on the table where the ball will land as it rolls off the chute and floats through the air. Place a sheet of paper on the marked place, and on it - a sheet of carbon paper with the “working” side down. Press down these sheets with a book so that they do not move during the experiments. measure distance from marked point to pointA . Lower the chute so that the distance from the bottom edge of the chute to the table is 10 cm, repeat the experiment.
After leaving the chute, the ball moves along a parabola, the top of which is at the point where the ball leaves the chute. Let's choose a coordinate system, as shown in the figure. Initial ball height and flight range related by the ratio According to this formula, with a decrease in the initial height by 4 times, the flight range decreases by 2 times. Having measured And you can find the speed of the ball at the moment of separation from the chute according to the formula 
