The relative velocity of an object is one that is measured with respect to a given observer, since another observer can obtain a different measurement. Velocity always depends on the observer who measures it.
Therefore, the speed of an object measured by a certain person will be the relative speed with respect to it. Another observer can obtain a different value for the velocity, even if it is the same object.
Since two observers A and B moving relative to each other can have different measurements of a third moving object P, it is necessary to look for a relationship between the positions and velocities of P seen by A and B.
Figure 1 shows two observers A and B with their respective reference systems, from which they measure the position and velocity of the object P.
Each observer A and B measures the position and velocity of the object P at a given instant of time t. In classical (or Galilean) relativity the time for observer A is the same as for observer B regardless of their relative velocities.
This article is about classical relativity that is valid and applicable to most everyday situations in which objects have speeds much slower than that of light..
We denote the position of observer B with respect to A as rBA. Since position is a vector quantity we use bold to indicate it. The position of the object P with respect to A is denoted as rPA and that of the same object P with respect to B rPB.
Article index
There is a vector relationship between these three positions that can be deduced from the representation of figure 1:
rPA= rPB + rBA
If the derivative of the previous expression is taken with respect to time t we will obtain the relationship between the relative velocities of each observer:
VPA= VPB + VBA
In the previous expression we have the relative speed of P with respect to A as a function of the relative speed of P with respect to B and the relative speed of B with respect to A.
Similarly, the relative velocity of P relative to B can be written as a function of the relative velocity of P relative to A and the relative velocity of A relative to B.
VPB= VPA + VAB
It should be noted that the relative velocity of A with respect to B is equal to and contrary to that of B with respect to A:
VAB = -VBA
A car goes on a straight road, which goes from west to east, with a speed of 80 km / h while in the opposite direction (and from the other lane) a motorcycle comes with a speed of 100 km / h.
A boy is traveling in the back seat of the car who wants to know the relative speed of a motorcycle that is approaching him. To find out the answer, the child will apply the relationships that he has just read in the previous section, identifying each coordinate system as follows:
-A is the coordinate system of an observer on the road and with respect to it the speeds of each vehicle have been measured.
-B is the car and P is the motorcycle.
If you want to calculate the speed of motorcycle P with respect to car B, the following relationship will be applied:
VPB= VPA + VAB=VPA - VBA
Taking the west-east direction as positive we have:
VPB= (-100 km / h - 80 km / h) i = -180 km / h i
This result is interpreted as follows: the motorcycle moves relative to the car with a speed of 180 km / h and direction -i, that is, from east to west.
The motorcycle and the car have crossed each other following their lane. The child in the back seat of the car sees the motorcycle move away and now wants to know how fast it is moving away from him, assuming that both the motorcycle and the car maintain the same speeds as before crossing..
To know the answer, the child applies the same relationship that was used previously:
VPB= VPA + VAB=V PA - VBA
VPB= -100 km / h i - 80 km / h i = -180 km / h i
And now the bike is moving away from the car with the same relative speed with which it was approaching before they crossed..
The same motorcycle from part 2 is returned maintaining its same speed of 100 km / h but changing its direction. That is, the car (which continues at a speed of 80 km / h) and the motorcycle both move in a positive east-west direction.
At a certain point, the motorcycle passes the car, and the child in the back seat of the car wants to know the relative speed of the motorcycle with respect to him when he sees it passing by..
To obtain the answer, the child applies the relations of relative motion again:
VPB= VPA + VAB=VPA - VBA
VPB= +100 km / h i - 80 km / h i = 20 km / h i
The child from the back seat watches the motorcycle overtaking the car at a speed of 20 km / h.
A motor boat crosses a river that is 600 m wide and flows from north to south. The speed of the river is 3 m / s. The speed of the boat with respect to the river water is 4 m / s towards the east.
(i) Find the speed of the boat relative to the river bank.
(ii) Indicate the speed and direction of the boat relative to land.
(iii) Calculate the crossover time.
(iv) How far it will have moved south with respect to the starting point.
There are two reference systems: the solidary reference system on the river bank that we will call 1 and the reference system 2, which is an observer floating on the river water. The object of study is boat B.
The speed of the boat relative to the river is written in vector form as follows:
VB2 = 4 i m / s
The speed of observer 2 (raft on the river) with respect to observer 1 (on land):
Vtwenty-one = -3 j m / s
You want to find the speed of the boat with respect to land VB1.
VB1 = VB2 + Vtwenty-one
VB1 = (4 i - 3 j) m / s
The speed of the boat will be the modulus of the previous speed:
|VB1| = (42 + (-3) 2) ½ = 5 m / s
And the address will be:
θ = arctan (-¾) = -36.87º
The boat crossing time is the quotient between the width of the river and the x component of the boat's speed with respect to land.
t = (600m) / (4 m / s) = 150 s
To calculate the drift that the boat had to the south, the y component of the boat's speed with respect to land is multiplied by the crossing time:
d = -3 j m / s * 150 s = -450 j m
The displacement towards the south with respect to the starting point is 450m.
Yet No Comments