Calculation of displacement in motion along a straight line
For simplicity we will analyze motion along the x axis of the coordinate system. The general formula for displacement of an object in motion with constant velocity is defined in Physics as
or, in the form of equation
If the object is moving with an acceleration the velocity is different at any moment of time and calculation of total displacement during the time of motion cannot be performed with Formula M1.16. In such a case we apply a special procedure by calculating the total displacement as a sum of a large number of displacements, each calculated for very a small period of time. One such displacement Δxi can be written as
where v(ti) is a velocity at given instance of time ti and Δt is a short lapse of time. During such a short time the velocity is nearly constant so we can use Formula M1.16 to calculate this small displacement. The approximate total displacement x will be equal to the sum of all displacements Δx created during the time of motion.
t1 is the moment of time when the motion starts, tn – the time the motion stops. To get the exact value of displacement we have to find the value of the right hand side of Equation M1.18 in the limiting case, when Δt → 0.
and this limit is known as an integral
Note that Δt → 0 in Equation M1.19 implies that number of intervals of time Δt becomes infinite - n → ∞.
The C is a constant of integration and can be found from the following reasoning. At time instant t=0, that is at the moment we start the calculation of the displacement, the already created displacement has some value x0. In other words, at the time instant we start calculation this object may have already traveled a distance x0, which is called the initial displacement.
Substituting at time t=0, x = x0, and ∫v(t)dt=0 , one gets
and the formula for displacement is now
It cannot be used directly to find the distance traveled by an object. We must know the explicit form of velocity v(t) as a function of time.
The simplest case is when velocity is constant, does not depend on time, v(t) = v. Substituting this constant velocity into Formula M1.21 we get
If you are not familiar with calculus and don’t know how to calculate integrals, just remember the final formula
The motion can be along any direction, not only along the x axis so this formula is usually written in the form.
which does not imply any specific direction of motion.
The next simplest case is when the velocity changes, but is a linear function of time
where a is acceleration. We do not use a vector notation for velocity and acceleration, because all the time we are discussing motion along a straight line. There is one direction of displacement, velocity, and acceleration. The Equation M1.25 tells us that an object starts from rest at time t=0 and gains the velocity uniformly with time. Substituting Equation M1.25 into M1.21 we get
Solving integral in this equation leads to the formula for the displacement
If you are not familiar with integration, don’t worry just remember formula M1.27, which may be used in solving problems.
A more complicated case is when at time t=0 an object already traveled distance x0 and has the velocity v0. From that moment (t=0) on it started moving with constant acceleration a. Velocities are additive (as long as they are much smaller than velocity of light = 300,000 km/s) therefore velocity on our object is given by
and after substituting it into Equation M1.21, we get the formula for displacement
which after solving the integral and rearranging reads
If you are not familiar with calculus, don’t pay attention to all the equations with integrals or derivatives. In solving most of the problems you will need only equations without these “fancy” symbols. But do not mistake the symbol of a derivative like with the operation of dividing dx by dt.
In general case one may want to calculate the displacement of an object moving with velocity given by an arbitrary function, for example
n is the frequency in cycles per second and the vertical lines denote the absolute value of the expression inserted between them.
Absolute value: |x| is defined as “unsigned” portion of x.
For example, |3| = 3, |-5| = 5.
The Formula M1.31 tells that velocity of an moving object changes from 0 to v0 periodically with frequency n. The calculation of displacement for such a motion requires the knowledge of calculus. You can find an example of such a calculation among the problems attached to this paragraph. You can smoothly go through this tutorial, even if you omit parts of material involving math you are not familiar with. So don’t worry when you see some strange equations or symbols.
Equations M1.23 and M1.27 are the special cases of Equation M1.30. Substituting into Equation M1.30:
Therefore the basic equation for solving most of the problems in Physics involving the calculation of displacement is Formula M1.30.
In more general cases one can have acceleration changing with time a = f(t) - a is a function of time. Such specific motions will be considered in some problems.