When mechanical motion is studied in physics, attention is drawn to two main quantities - this is speed and acceleration. Knowing their dependence on time allows you to determine the coordinate of the body in space at any time. In this article, we present all acceleration formulas for its various types.
General concept
Acceleration is a physical quantity. It determines how fast the speed changes. How to find acceleration? The formula is below:
a = dv / dt
That is, the quantity a is the time derivative of the velocity. This expression allows you to calculate the so-called full instantaneous acceleration, that is , it characterizes the value at a given specific moment in time.
In practice, it is often important to know not instantaneous, but some average acceleration with which the body moved for a certain time. It can be calculated using the following formula:
a = (v 2 -v 1 ) / (t 2 -t 1 ).
Here v 2 and v 1 - instantaneous speeds at time t 2 and t 1 respectively.
Curved motion
Acceleration determines the magnitude of the change in speed. The latter varies more than one parameter. It can vary both in direction and in size. If the body moves in a straight line, then the velocity vector retains its direction. For such a movement, the total acceleration is determined solely by a change in the velocity modulus. It is called tangent or tangential.
If the body moves along an arbitrary curved path, then the velocity vector necessarily changes. This fact leads to the appearance of a normal acceleration component. Let's consider in more detail. Opening the topic of all acceleration formulas, we give an expression for calculating normal acceleration:
a n = v 2 / r
This equality allows us to draw two important conclusions:
- Firstly, the normal component depends on the speed module, and not on its change, as the tangential component.
- Secondly, it is inversely proportional to the radius of curvature of the trajectory r. When the body rotates uniformly around the circumference, there is only a normal component.
To determine the absolute acceleration modulus, one should use the following formula:
a = β (a n 2 + a t 2 )
Vector a is defined as the sum of vectors a n and a t , the first of them is directed perpendicular to the path to the center of its curvature, and the second - along the tangent to the path in the direction of change of the velocity modulus.
Free fall
So they call the vertical movement of the body in the gravitational field of the planet, carried out under the influence of gravity. Typically, the corresponding acceleration is denoted by the letter g. For example, for the Earth it is 9.81 m / s 2 . We give all the formulas for accelerating g:
g = G * M / R 2 ;
g = F / m;
g = 2 * h / t 2
The first of these expressions allows you to determine the acceleration g, if the mass of the planet M and its radius R. are known. G is the gravitational constant. This formula follows from Newton's law of gravity.
The second expression is the well-known equation for gravity F, which acts on a body of mass m.
Finally, the third formula determines the acceleration through the fall height h and the fall time t of the body without initial velocity. This expression is one of the basic kinematics of rectilinear motion.
Angular acceleration
This view also cannot be ignored. Given all the acceleration formulas, it is worth noting that the angular value is convenient to use when the system rotates around a certain axis. It determines the rate of change of angular velocity and is expressed in radians per square second. To determine this type of acceleration, the following formulas are used:
Ξ± = dΟ / dt = d 2 ΞΈ / dt 2 ;
Ξ± = M / I
The first equality shows that to determine the angular acceleration Ξ±, one should find the derivative of the angular velocity Ο with respect to time or the second derivative with respect to time of the angle of rotation ΞΈ.
The second line is an expression that follows from the equation of moments. Here M is the moment of force that unwinds the system, I is the moment of inertia, which plays the role of body mass during linear motion.