How to write the equation of uniformly accelerated motion using a graph. Uniformly accelerated motion. Path and displacement in rectilinear uniformly accelerated motion

Questions.

1. Write down the formula by which you can calculate the projection of the instantaneous velocity vector of rectilinear uniformly accelerated motion if you know: a) the projection of the initial velocity vector and the projection of the acceleration vector; b) projection of the acceleration vector given that the initial speed is zero.

2. What is the projection graph of the velocity vector of uniformly accelerated motion at an initial speed: a) equal to zero; b) not equal to zero?

3. How are the movements, the graphs of which are presented in Figures 11 and 12, similar and different from each other?

In both cases, the movement occurs with acceleration, but in the first case the acceleration is positive, and in the second case it is negative.

Exercises.

1. A hockey player lightly hit the puck with his stick, giving it a speed of 2 m/s. What will be the speed of the puck 4 s after impact if, as a result of friction with ice, it moves with an acceleration of 0.25 m/s 2?

2. A skier slides down a mountain from a state of rest with an acceleration equal to 0.2 m/s 2 . After what period of time will its speed increase to 2 m/s?

3. In the same coordinate axes, construct graphs of the projection of the velocity vector (on the X axis, codirectional with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox = 1 m/s, a x = 0.5 m/s 2 ; b) v ox = 1 m/s, a x = 1 m/s 2; c) v ox = 2 m/s, a x = 1 m/s 2.
The scale is the same in all cases: 1 cm - 1 m/s; 1cm - 1s.

4. In the same coordinate axes, construct graphs of the projection of the velocity vector (on the X axis, codirectional with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox = 4.5 m/s, a x = -1.5 m/s 2 ; b) v ox = 3 m/s, a x = -1 m/s 2
Choose the scale yourself.

5. Figure 13 shows graphs of the velocity vector modulus versus time for the rectilinear motion of two bodies. With what absolute acceleration does body I move? body II?

Uniformly accelerated motion is a motion in which the acceleration vector does not change in magnitude and direction. Examples of such movement: a bicycle rolling down a hill; a stone thrown at an angle to the horizontal. Uniform motion is a special case of uniformly accelerated motion with acceleration equal to zero.

Let us consider the case of free fall (a body thrown at an angle to the horizontal) in more detail. Such movement can be represented as the sum of movements relative to the vertical and horizontal axes.

At any point of the trajectory, the body is affected by the acceleration of gravity g →, which does not change in magnitude and is always directed in one direction.

Along the X axis the movement is uniform and rectilinear, and along the Y axis it is uniformly accelerated and rectilinear. We will consider the projections of the velocity and acceleration vectors on the axis.

Formula for speed during uniformly accelerated motion:

Here v 0 is the initial velocity of the body, a = c o n s t is the acceleration.

Let us show on the graph that with uniformly accelerated motion the dependence v (t) has the form of a straight line.

Acceleration can be determined by the slope of the velocity graph. In the figure above, the acceleration modulus is equal to the ratio of the sides of triangle ABC.

a = v - v 0 t = B C A C

The larger the angle β, the greater the slope (steepness) of the graph relative to the time axis. Accordingly, the greater the acceleration of the body.

For the first graph: v 0 = - 2 m s; a = 0.5 m s 2.

For the second graph: v 0 = 3 m s; a = - 1 3 m s 2 .

Using this graph, you can also calculate the displacement of the body during time t. How to do it?

Let us highlight a small period of time ∆ t on the graph. We will assume that it is so small that the movement during the time ∆t can be considered a uniform movement with a speed equal to the speed of the body in the middle of the interval ∆t. Then, the displacement ∆ s during the time ∆ t will be equal to ∆ s = v ∆ t.

Let us divide the entire time t into infinitesimal intervals ∆ t. The displacement s during time t is equal to the area of the trapezoid O D E F .

s = O D + E F 2 O F = v 0 + v 2 t = 2 v 0 + (v - v 0) 2 t.

We know that v - v 0 = a t, so the final formula for moving the body will take the form:

s = v 0 t + a t 2 2

In order to find the coordinate of the body at a given time, you need to add displacement to the initial coordinate of the body. The change in coordinates depending on time expresses the law of uniformly accelerated motion.

Law of uniformly accelerated motion

y = y 0 + v 0 t + a t 2 2 .

Another common kinematics problem that arises when analyzing uniformly accelerated motion is finding the coordinate for given values of the initial and final velocities and acceleration.

Eliminating t from the equations written above and solving them, we obtain:

s = v 2 - v 0 2 2 a.

From the known initial speed, acceleration and displacement, you can find the final speed of the body:

v = v 0 2 + 2 a s .

For v 0 = 0 s = v 2 2 a and v = 2 a s

Important!

The quantities v, v 0, a, y 0, s included in the expressions are algebraic quantities. Depending on the nature of the movement and the direction of the coordinate axes under the conditions of a specific task, they can take on both positive and negative values.

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Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. Designate

Uniform motion graphs

Dependence of acceleration on time. Since during uniform motion the acceleration is zero, the dependence a(t) is a straight line that lies on the time axis.

Dependence of speed on time. The speed does not change over time, the graph v(t) is a straight line parallel to the time axis.

The numerical value of the displacement (path) is the area of the rectangle under the speed graph.

Dependence of the path on time. Graph s(t) - sloping line.

Rule for determining speed using graph s(t): The tangent of the angle of inclination of the graph to the time axis is equal to the speed of movement.

Graphs of uniformly accelerated motion

Dependence of acceleration on time. Acceleration does not change with time, has a constant value, the graph a(t) is a straight line parallel to the time axis.

Dependence of speed on time. With uniform motion, the path changes according to a linear relationship. In coordinates. The graph is a sloping line.

The rule for determining the path using the graph v(t): The path of a body is the area of the triangle (or trapezoid) under the velocity graph.

The rule for determining acceleration using the graph v(t): The acceleration of a body is the tangent of the angle of inclination of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.

Dependence of the path on time. During uniformly accelerated motion, the path changes according to