# 2.4 Displacement-time graphs

A particle's position, *x*, is always measured from the origin of the coordinate system. However, in describing real motions it is often important to know where something is located relative to a point other than the origin. Figure 10 shows a case in point; in a high-speed pursuit neither the police nor the robbers are likely to be very interested in their location relative to the origin, but both will be interested in the location of their own vehicle relative to the other. The physical quantity used to describe the location of one point relative to another is called **displacement**. In the case of Figure 10, the displacement of the getaway car from the police car is 400 m and the displacement of the police car from the getaway car is −400 m. In each case the displacement of a body is determined by subtracting the position coordinate of the reference body from the position coordinate of the body of interest. Thus the displacement (measured along the *x*-axis) of a particle with position *x* from a chosen reference point with position *x*_{ref} is given by

Notice that displacements, like positions, may be positive or negative depending on the direction of the displacement. Thus *s _{x}* is always positive when

*x*is to the right of

*x*

_{ref}and is negative when

*x*is to the left of

*x*

_{ref}. Also note that since displacements are measured along a definite axis it makes sense to represent them by a symbol,

*s*, that includes a reference to that axis. This symbol may be read as '

_{x}*s*subscript

*x*' or, more simply, as '

*s*sub

*x*'.

A special case of Equation 1 is when the reference point *x*_{ref} is the origin. Then *x*_{ref} = 0 and *s _{x}* =

*x*. Thus the displacement of a point from the origin is the position coordinate of the point. Once we know the displacement

*s*of one object from another, it is easy to work out the

_{x}**distance**

*s*between those two objects. The distance is just the numerical value of the displacement, including the unit of measurement but ignoring any overall negative sign. So, when the displacement of the police car from the getaway car is

*s*= −400 m, the distance between them is

_{x}*s*= 400 m. We describe this simple relationship between distance and displacement by saying that the distance between two objects is given by the

**magnitude**of the displacement of one from the other, and we indicate it mathematically by writing

The two vertical bars | | constitute a **modulus sign** and indicate that you should work out the value of whatever they enclose and then take its magnitude (i.e. ignore any overall minus sign).

## Question 4

In Figure 8, what is the displacement *s _{x}* of the car from the pedestrian, and what is the distance

*s*between them?

### Answer

*s _{x}* = 50 m, and

*s*= 50 m.

In many circumstances it is more valuable to plot a **displacement-time graph** rather than a position-time graph. In order to do this you either have to know the relevant displacements at various times, or you need to know enough about the positions of both the bodies involved to work out the displacements. Table 3 contains some plausible data about the positions of the two cars in Figure 10; use it to answer the following question.

## **Table 3:** The position coordinates of the police car *x*_{pol} and getaway car *x*_{get} at various times *t*

t/s | x_{pol}/m | x_{get}/m |
---|---|---|

0 | −200 | 200 |

5 | −115 | 237 |

10 | −45 | 265 |

15 | 10 | 293 |

20 | 62 | 317 |

25 | 108 | 337 |

30 | 149 | 351 |

35 | 184 | 364 |

40 | 213 | 377 |

45 | 239 | 388 |

50 | 263 | 394 |

55 | 283 | 399 |

60 | 300 | 400 |

## Question 5

Plot a graph to show how the displacement of the getaway car from the police car depends on time.

### Answer

Since we are interested in the displacement of the getaway car from the police car, we need to plot *x*_{get} − *x*_{pol}. You may find it useful to tabulate these values by adding another column (headed *x*_{get} − *x*_{pol}) to Table 3. The result of the plot is given in Figure 11.