The study of classical mechanical motion in physics is engaged in kinematics. Unlike dynamics, science studies why bodies move. She answers the question of how they do it. In this article, we consider what acceleration and motion with constant acceleration are.
The concept of acceleration
When a body moves in space, for some time it overcomes a certain path, which is the length of the trajectory. To calculate this path, use the concepts of speed and acceleration.
Speed as a physical quantity characterizes the speed in time of a change in the distance traveled. The speed is directed tangentially to the trajectory in the direction of movement of the body.
Acceleration is a slightly more complex quantity. In short, it describes the change in speed at a given point in time. The mathematical definition of acceleration looks like this:
a¯ = dv¯ / dt.
To understand this formula more clearly, we give a simple example: suppose that in 1 second of movement the body’s speed increased by 1 m / s. These figures, substituted in the expression above, lead to the result: the acceleration of the body during this second was equal to 1 m / s 2 .
The direction of acceleration is completely independent of the direction of speed. Its vector coincides with the vector of the resulting force that causes this acceleration.
An important point should be noted in the above definition of acceleration. This value characterizes not only the change in speed modulo, but also in direction. The latter fact should be taken into account in the case of curvilinear motion. Further in the article only rectilinear motion will be considered.
Constant Acceleration Speed
Acceleration is constant if it maintains its modulus and direction during movement. Such a movement is called uniformly accelerated or equally slow - it all depends on whether the acceleration leads to an increase in speed or to a decrease in it.
In the case of a body moving with constant acceleration, you can determine the speed using one of the following formulas:
v = a * t;
v = v 0 + a * t;
v = v 0 - a * t.
The first two equations characterize uniformly accelerated displacement. The difference between them is that the second expression is applicable for the case of a nonzero initial velocity.
The third equation is an expression for speed with equally slow motion with constant acceleration. The acceleration is directed against the speed.
The graphs of all three functions v (t) are straight lines. In the first two cases, the lines have a positive slope relative to the abscissa axis, in the third case this slope is negative.
Formulas traveled
For the path in the case of motion with acceleration constant (acceleration a = const), it is not difficult to obtain formulas if we calculate the integral of speed over time. Having performed this mathematical operation for the above three equations, we obtain the following expressions for the path L:
L = a * t 2/2;
L = v 0 * t + a * t 2/2;
L = v 0 * t - a * t 2/2.
The graphs of all three functions of the path from time to time are parabolas. In the first two cases, the right branch of the parabola increases, and for the third function, it gradually reaches a certain constant, which corresponds to the distance traveled until the body stops completely.
The solution of the problem
Moving at a speed of 30 km / h, the car began to accelerate. In 30 seconds, he walked a distance of 600 meters. What was the acceleration of the car equal to?
First of all, we translate the initial speed from km / h to m / s:
v 0 = 30 km / h = 30000/3600 = 8.333 m / s.
Now we write the equation of motion:
L = v 0 * t + a * t 2/2.
From this equality we express acceleration, we obtain:
a = 2 * (L - v 0 * t) / t 2 .
All physical quantities in this equation are known from the conditions of the problem. We substitute them into the formula and get the answer: a ≈ 0.78 m / s 2 . Thus, moving with acceleration constant, the car increased its speed by 0.78 m / s for every second.
We will also calculate (for interest) what speed he gained after 30 seconds of accelerated movement, we get:
v = v 0 + a * t = 8.333 + 0.78 * 30 = 31.733 m / s.
The resulting speed is 114.2 km / h.