We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist was moving uniformly, his equation of motion is:
(the origin of coordinates is placed at the starting point, so the initial coordinate of the cyclist is zero).
The motorcyclist was moving at a uniform speed. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also equal to zero (the motorcyclist began to move from a state of rest).
Considering that the motorcyclist started moving a little later, the motorcyclist's equation of motion is:
In this case, the speed of the motorcyclist changed according to the law:
At the moment when the motorcyclist caught up with the cyclist, their coordinates are equal, i.e. or:
Solving this equation with respect to , we find the meeting time:
This is a quadratic equation. We define the discriminant:
Define roots:
Substitute the numerical values into the formulas and calculate:
We discard the second root as not corresponding to the physical conditions of the problem: the motorcyclist could not catch up with the cyclist 0.37 s after the cyclist began to move, since he himself left the starting point only 2 s after the cyclist started.
Thus, the time when the motorcyclist caught up with the cyclist:
Substitute this value of time into the formula for the law of change in the speed of a motorcyclist and find the value of his speed at this moment:
2) Graphical way.
On the same coordinate plane, we build graphs of changes in the coordinates of the cyclist and motorcyclist over time (the graph for the coordinates of the cyclist is in red, for the motorcyclist - in green). It can be seen that the dependence of the coordinate on time for a cyclist is a linear function, and the graph of this function is a straight line (the case of uniform rectilinear motion). The motorcyclist was moving with uniform acceleration, so the dependence of the motorcyclist's coordinates on time is a quadratic function, the graph of which is a parabola.
Formulas for the rectilinear motion of a material point are derived for three ways of specifying the motion - with a known dependence of the coordinate on time; with a known dependence of acceleration on time and acceleration on coordinates. Rectilinear uniform and rectilinear uniformly accelerated motions are considered.
ContentBasic formulas for rectilinear motion
Let the material point move along the axis. Further, and denote the coordinate and velocity of the point at the initial moment of time .
If the law of change of its coordinates from time is given:
,
then differentiating the coordinate with respect to time, we obtain the speed and acceleration of the point:
;
.
Let us the dependence of acceleration on time is known:
.
Then the dependences of the speed and coordinates on time are determined by the formulas:
(1)
;
(2)
;
(3)
;
(4)
.
Let us the dependence of the acceleration on the coordinate is known:
.
Then the dependence of the velocity on the coordinate has the form:
(5)
.
The dependence of the coordinate on time is defined implicitly:
(6)
.
For rectilinear uniform motion:
;
;
.
For rectilinear uniformly accelerated motion:
;
;
;
.
The formulas given here can be applied not only to rectilinear motion, but also for some cases of curvilinear motion. For example, for three-dimensional movement in a rectangular coordinate system, if the movement along the axis does not depend on the projections of quantities on other coordinate axes. Then formulas (1) - (6) give dependencies for the projections of quantities onto the axis .
Also, these formulas are applicable when moving along a given trajectory with a natural way of setting motion. Only here, the length of the arc of the trajectory, measured from the selected reference point, acts as a coordinate. Then instead of the projections and one should substitute and - the projections of the velocity and acceleration on the chosen direction of the tangent to the trajectory.
Rectilinear motion with a known dependence of the coordinate on time
Consider the case when a material point moves in a straight line. We choose a coordinate system with the origin at an arbitrary point . Let's direct the axis along the line of movement of the point. Then the position of the point is uniquely determined by the value of one coordinate .
If the law of coordinate change from time is given:
,
then differentiating with respect to time , we find the law of speed change:
.
When the point moves in the positive direction of the axis (left to right in the figure). When the point moves in the negative direction of the axis (right to left in the figure).
Differentiating the speed with respect to time, we find the law of change of acceleration:
.
Since the straight line has no curvature, the radius of curvature of the trajectory can be considered infinitely large, . Then the normal acceleration is zero:
.
That is, the acceleration of the point is tangential (tangential):
.
Which is quite natural, since both the speed and acceleration of the point are directed tangentially to the trajectory - the straight line along which the movement occurs.
If and of the same sign (that is, both are positive or both are negative), then the speed modulus increases (the speed increases in absolute value). If and of different signs, then the speed modulus decreases (the speed decreases in absolute value).
Rectilinear motion with known acceleration
Time Dependent Acceleration
Let us know the law of change of acceleration with time:
.
Our task is to find the law of change of speed and the law of change of coordinates from time:
;
.
Let's apply the formula:
.
This is a first-order differential equation with separable variables
;
.
Here is the constant of integration. This shows that only by the known dependence of acceleration on time, it is impossible to unambiguously determine the dependence of speed on time. We have obtained a whole set of speed change laws that differ from each other by an arbitrary constant . To find the law of change of speed we need, we must specify one more value. As a rule, this value is the value of the speed at the initial moment of time . To do this, we pass from the indefinite integral to the definite one:
.
Let be the speed of the point at the initial moment of time . Substitute :
;
;
.
Thus, the law of change of speed with time has the form:
(1)
.
Similarly, we define the law of change of coordinates from time.
.
(2)
.
Here - the value of the coordinate at the initial moment of time .
We substitute (1) into (2).
.
Domain of integration in the double integral.
If we change the order of integration in the double integral, we get:
.
Thus, we have received the following formulas:
(3)
;
(4)
.
Coordinate dependent acceleration
Let now we know the law of change of acceleration from the coordinate:
.
We need to solve the differential equation:
.
This differential equation does not explicitly contain the independent variable. The general method for solving such equations is discussed on the page “Higher-order differential equations that do not contain an explicit independent variable”. According to this method, we consider that is a function of :
;
.
We separate the variables and integrate:
;
;
;
.
When extracting the root, one must take into account that the speed can be both positive and negative. At a small distance from the point , the sign is determined by the sign of the constant . However, if the acceleration is directed opposite to the speed, then the speed of the point will decrease to zero and the direction of motion will change to the opposite. Therefore, the correct sign, plus or minus, is chosen when considering a particular movement.
(5)
.
At the beginning of the movement
.
Now we determine the dependence of the coordinate on time. The differential equation for the coordinate is:
.
This is a separable differential equation. We separate the variables and integrate:
(6)
.
This equation defines the dependence of the coordinate on time in an implicit form.
Rectilinear uniform motion
Let us apply the results obtained above for the case of rectilinear uniform motion. In this case, the acceleration
.
;
. That is, the speed is constant, and the coordinate depends linearly on time. Formulas (5) and (6) give the same result.
Rectilinear uniformly accelerated motion
Now consider rectilinear uniformly accelerated motion.
In this case, the acceleration is a constant value:
.
According to formulas (1) and (2) we find:
;
.
If we apply formula (5), then we obtain the dependence of the velocity on the coordinate:
.
Rectilinear motion in vector form
The resulting formulas can be represented in vector form. To do this, it suffices to multiply the equations that determine and by the unit vector (ort) directed along the axis.
Then the radius vector of the point, the velocity and acceleration vectors have the form:
;
;
.
What is uniformly accelerated motion?
Uniformly accelerated motion in physics is such a motion, the acceleration vector of which does not change in magnitude and direction. In simple terms, uniformly accelerated motion is an uneven motion (that is, going at different speeds), the acceleration of which is constant over a certain period of time. Imagine that starts moving, for the first 2 seconds its speed is 10 m / s, the next 2 seconds it is already moving at a speed of 20 m / s, and after another 2 seconds already at a speed of 30 m / s. That is, every 2 seconds it accelerates by 10 m / s, such a movement is uniformly accelerated.
From this we can derive an extremely simple definition of uniformly accelerated motion: this is the motion of any physical body, in which its speed changes in the same way for equal periods of time.
Examples of uniformly accelerated motion
A clear example of uniformly accelerated motion in everyday life can be a bicycle going downhill (but not a bicycle driven by a cyclist), or a stone thrown at a certain angle to the horizon.
By the way, the example with a stone can be considered in more detail. At any point of the flight trajectory, the free fall acceleration g acts on the stone. The acceleration g does not change, that is, it remains constant and is always directed in one direction (in fact, this is the main condition for uniformly accelerated motion).
It is convenient to represent the flight of a thrown stone as a sum of movements relative to the vertical and horizontal axes of the coordinate system.
If along the X axis the movement of the stone will be uniform and rectilinear, then along the Y axis it will be uniformly accelerated and rectilinear.
Formula of uniformly accelerated motion
The formula for speed in uniformly accelerated motion will look like this:
Where V 0 is the initial speed of the body, and is the acceleration (as we remember, this value is a constant), t is the total flight time of the stone.
With uniformly accelerated motion, the dependence V(t) will look like a straight line.
Acceleration can be determined from the slope of the velocity graph. In this figure, it is equal to the ratio of the sides of the triangle ABC.
The larger the angle β, the greater the slope and, as a result, the steepness of the graph with respect to the time axis, and the greater will be the acceleration of the body.
- Sivukhin DV General course of physics. - M.: Fizmatlit, 2005. - T. I. Mechanics. - S. 37. - 560 p. - ISBN 5-9221-0225-7.
- Targ S. M. A short course in theoretical mechanics. - 11th ed. - M .: "Higher School", 1995. - S. 214. - 416 p. - ISBN 5-06-003117-9.
Uniformly accelerated motion, video
Graphical representation of uniform rectilinear motion
Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. Designate:
V (t) - change in speed with time
a(t) - change in acceleration with time
Behind acceleration versus time. Since the acceleration is equal to zero during uniform motion, the dependence a(t) is a straight line that lies on the time axis.
Speed versus time. Since the body moves in a straight line and uniformly (v = const ), i.e. speed does not change with time, then the graph with the dependence of speed on time v(t) is a straight line parallel to the time axis.
The projection of the body displacement is numerically equal to the area of the rectangle AOBC under the graph, since the magnitude of the displacement vector is equal to the product of the velocity vector and the time during which the movement was made.
The rule for determining the path according to the schedule v(t): with rectilinear uniform motion, the modulus of the displacement vector is equal to the area of the rectangle under the velocity graph.
Dependence of displacement on time. Graph s(t) - sloping line :
It can be seen from the graph that the velocity projection is equal to:
Having considered this formula, we can say that the larger the angle, the faster the body moves and it travels a greater distance in less time.
The rule for determining the speed according to the schedule s(t): The tangent of the slope of the graph to the time axis is equal to the speed of movement.
Uneven linear motion.
Uniform motion is motion at a constant speed. If the speed of a body changes, it is said that it is moving unevenly.
A movement in which a body makes unequal movements in equal intervals of time is called uneven or variable motion.
To characterize non-uniform motion, the concept of average speed is introduced.
Average moving speed is equal to the ratio of the entire path traveled by a material point to the time interval for which this path has been traveled.
In physics, the greatest interest is not the average, but instantaneous speed , which is defined as the limit to which the average speed tends over an infinitesimal time interval Δ t:
instantaneous speedvariable motion is called the speed of the body at a given time or at a given point in the trajectory.
The instantaneous velocity of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at that point.
The difference between average and instantaneous speeds is shown in the figure.
The movement of a body, in which its speed for any equal intervals of time changes in the same way, is called uniformly accelerated or uniform motion.
Acceleration -it is a vector physical quantity that characterizes the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.
If the speed changes the same throughout the entire time of movement, then the acceleration can be calculated by the formula:
Designations:
V x - The speed of the body with uniformly accelerated motion in a straight line
V x o - Initial speed of the body
a x - Acceleration of the body
t - body movement time
Acceleration shows how quickly the speed of a body changes. If the acceleration is positive, then the speed of the body increases, the movement is accelerated. If the acceleration is negative, then the speed is decreasing, the movement is slow.
Unit of acceleration in SI [m/s 2 ].
Acceleration is measured accelerometer
Speed equation for uniformly accelerated motion: v x = v xo + a x t
The equation of uniformly accelerated rectilinear motion(displacement with uniformly accelerated motion):
Designations:
S x - Movement of the body with uniformly accelerated motion in a straight line
V x o - Initial speed of the body
V x - The speed of the body with uniformly accelerated motion in a straight line
a x - Acceleration of the body
t - body movement time
More formulas for finding displacement during uniformly accelerated rectilinear motion, which can be used in solving problems:
If the initial, final velocities and acceleration are known.
If the initial, final speeds of movement and the time of the entire movement are known
Graphical representation of non-uniform rectilinear motion
Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. Designate:
V(t) - speed change with time
S(t) - change in displacement (path) over time