Acceleration in physics is one of the important kinematic characteristics, without the knowledge of which it is impossible to describe any kind of motion. In this article, we consider what this quantity is, and also give acceleration formulas for uniformly accelerated rectilinear motion.
Acceleration and the reason for its appearance
In physics, the value that characterizes the change in speed over time is called acceleration. The mathematical formula for acceleration looks like this:
a¯ = dv¯ / dt.
The faster the speed changes, the greater will be the acceleration of the body. For example, a = 1 m / s 2 means that in 1 second the speed increased by 1 m / s.
The acceleration of bodies arises due to the action of external forces of any nature on them. This fact was established by Newton in the XVII century. Currently, it is called Newton's 2nd law:
F¯ = m * a¯.
Both formulas indicate that the acceleration vector is directed towards the change in the velocity vector or towards the force vector (F¯ and dv¯ are directed identically). If the directions of the vectors a¯ and v¯ coincide, then the body will accelerate, if they are opposite, then the body will slow down its movement, if they are directed at a certain angle, then the trajectory of movement will be a curved line.
Equally accelerated rectilinear motion. Speed and acceleration
The indicated type of motion assumes that the trajectory of the body is a straight line, and the magnitude of the acceleration in the process of moving the body does not change either modulo or direction. Since the body moves in a straight line, the vectors a¯ and v¯ are directed either in one direction or in opposite directions.
Suppose the body was at rest. Then a constant force began to act on him, which gave him acceleration. In this case, the speed v at any time t can be calculated as follows:
v = a * t.
Where a = const. The graph of this equation is an increasing line that starts from a point (v = 0; t = 0).
If the body already had a certain speed v 0 before the action of the force, then the following formulas will be valid:
v = v 0 + a * t;
v = v 0 - a * t.
In the first case, we are talking about accelerated motion, in the second - about slow motion (braking).
From the last two expressions, one can obtain acceleration formulas with uniformly accelerated body motion in a straight line:
a = (vv 0 ) / t;
a = (v 0 -v) / t.
Time t is counted from the moment the force acts on the body.
Acceleration and path
When solving problems of uniformly accelerated movement, it is often necessary to find acceleration, knowing the distance traveled. We show which formulas should be used for this.
The path is easy to calculate with uniformly accelerated movement in a straight line. To do this, take the time integral of the equation v (t). Having performed this mathematical action, we obtain three working formulas:
S = a * t 2 ;
S = v 0 * t + a * t 2 ;
S = v 0 * t - a * t 2 .
The first expression describes the accelerated movement of the body from a state of rest, the second describes accelerated movement with the presence of an initial velocity, and the third describes braking. The graphs of all three functions S (t) are parabolas.
How to express acceleration from formulas of uniformly accelerated motion for a path? For this, it is necessary to isolate the factor a * t 2 in one part of the equality, and then divide all the equality by the square of time. From the formulas above we get:
a = S / t 2 ;
a = (S - v 0 * t) / t 2 ;
a = (v 0 * t - S) / t 2 .
The first equation is used to experimentally determine the acceleration of gravity g, when heavy bodies are thrown down from a certain height. Galileo conducted similar experiments at the end of the 16th century. Currently, absolute gravimeters are used to determine the acceleration g in the study area, the principle of which is also based on free fall.
The last two equations differ from each other only in the sign of acceleration. When braking, acceleration is considered negative.
All three expressions lead to the same unit of measurement for a - m / s 2 .
Acceleration calculation task
Having dealt with the basic acceleration formulas for uniformly accelerated movement, we will solve the following practical problem: the driver of the car, which was moving at a speed of 63 km / h, saw that a red traffic signal lit up ahead. After depressing the brake pedal, the car stopped completely after 100 meters. Knowing that the braking time took 14 seconds, it is necessary to calculate the corresponding acceleration.
To solve the problem, you can immediately use one of the above formulas:
a = (v 0 * t - S) / t 2 .
We translate the initial vehicle speed from km / h to m / s, we get:
v 0 = 63 * 1000/3600 = 17.5 m / s 2 .
Now you can substitute the values from the conditions of the problem and get the answer: a = 0.74 m / s 2 . Since we are talking about braking, this acceleration should be taken with a minus sign: a = - 0.74 m / s 2 .