One of the simple types of mechanical movement in the space of bodies is uniformly accelerated motion. It is described by certain kinematic formulas. In this article, we consider what the equation of speed is when moving uniformly accelerated.
The concept of speed and acceleration in physics
Before writing the equation of speed with uniformly accelerated body motion, we consider both physical quantities and their meaning.
Speed is a kinematic characteristic that determines the rate of change in the spatial coordinates of a body during its movement. The mathematical definition of speed looks like this:
v¯ = dl¯ / dt
Where dl¯ is the vector traveled during the dt path.
Speed is measured in m / s (meters per second). Its vector along the tangent is directed to the point of the trajectory at which the moving body is at a given moment in time.
Acceleration is the time derivative of speed. Acceleration shows how quickly the speed of the body changes, that is:
a¯ = dv¯ / dt
The value of a¯ is measured in m / s 2 (meters per square second). The direction of acceleration coincides with the difference of the velocity vectors. If we recall Newton’s law on the relationship between force and acceleration, we can establish that the vector a¯ always coincides with the vector of the resulting external force acting on the body.
What movement is called uniformly accelerated?
Now we know what speed and acceleration are. The equation of uniformly accelerated motion can be written down if you know what this type of movement of bodies is. The movement of the body will be uniformly accelerated only when its acceleration is constant for some time. By the constancy of acceleration we mean the immutability of the modulus and the vector of a¯.
The concept of uniformly accelerated motion is closely related to the concept of trajectory. If the trajectory is a straight line, then constant acceleration can be directed either along the velocity vector or against it. In the latter case, inhibition of the body will occur.
If the trajectory is a circle (rotation of bodies around a fixed axis), then uniformly accelerated motion assumes a constant angular acceleration. The latter is linearly related to the tangential component of full acceleration. In the case of uniform movement around the circumference, the total acceleration is not equal to zero, since there is a non-zero normal component of it.
Next, we consider the equations of speed when moving uniformly accelerated, taking into account the rectilinear trajectory.
Acceleration Equations
Let us carry out the following thought experiment. Suppose the car is at rest on the road. Then it begins to move, and in time t its speed becomes equal to v. Since the speed has changed from zero to v, the following expression can be written to accelerate a:
a = (v-0) / t =>
v = a * t
Thus, the product of constant acceleration by the time of motion will give the value of speed.
Now suppose the car picks up some speed v 0 and starts to slow down. In this case, the velocity equation with uniformly accelerated motion has the form:
v = v 0 - a * t
A minus sign indicates that the acceleration vector is directed against speed and seeks to reduce its module (the car stops).
Finally, if the vehicle already had some speed v 0 , and then the driver pressed the gas pedal, then you can calculate the value of v at any time t using the following formula:
v = v 0 + a * t
All three written equations in graphical form are straight lines. The graph of the first equation passes through the origin (t = 0; v = 0). The graphs of the second and third equations pass through the point (t = 0; v 0 ), while the graph of the second equation decreases, that is, it has a negative slope coefficient (-a), and the graph of the third increases (+ a).
Problem solving example
It is known that the car was moving at a speed of 70 km / h. After depressing the brake pedal, he began to stop. It is known that the acceleration of braking of the vehicle was equal to 3 m / s 2 . How long after pressing the brake pedal does the car stop completely?
In accordance with the condition of the problem, it is obvious that we need to apply the following equation of speed through acceleration to solve it:
v = v 0 - a * t
Since the vehicle stopped completely, its final speed v became equal to zero. This fact allows us to express the value of t from the equation written above, we have:
t = v 0 / a
A speed of 70 km / h corresponds to a value of 19.44 m / s. Substituting the value of the acceleration of braking, we come to the answer: t = 6.48 seconds.