Speed, Distance and Time

Speed, Distance and Time - Important Formula, Play Online Quiz

Relation between Distance Time and speed

  1. $speed = \dfrac{distance}{time}$

  2. $distance =speed × time$

  3. $time = \dfrac{distance}{speed}$

Average Speed

  • Let a person cover a certain distance with the speed of $x$ $km.$/$h.$ and the same distance with the speed of $y$ $km$/$h.$Then Average speed of whole journey is

$$ =\frac{2xy}{(x+y)} km \mathbin{/} h $$

  • If Three equal distances are covered by different speeds x, y and z km/h respectively,then average speed for the whole journey is $$=\left[\frac{3xyz}{xy+yz+zx}\right] \; km/h$$
  • If a certain distance is covered with speed of x km/h and another distance with a speed of y km/h but time interval for both journey being same,then average speed for whole journey is $$=\left[\frac{x+y}{2}\right]\;km/h$$
  • If a certain distance is covered with speed of x, y and z km/h respectively but time interval for the three journeys being equal then average speed is $$=\left[\frac{x+y+z}{3}\right]\;km/h$$
  • If a body covers some part of the journey at speed P and the remaining part of the journey at speed Q and the distance of the two parts of the journey are in the ratio M : N, then the average speed for the entire journey is $$=\left[\frac{M + N}{MQ+NP}\right]$$

  • If the ratio of the speeds of A and B is a : b, then the ratio of time taken by them to cover the same distance is $$=b : a$$

Kilometer/hour and meter/second conversion : -

$$x\;km/hrs= (x\times\frac{5}{18})\,m/s$$ $$x\;m/s=(x\times\frac{18}{5})\,km/h$$

Maths Short Tricks Formula

Type : 1

  • If a certain distance is covered with S1 speed in T1 time and with S2 speed in T2 time, then
    S1T1 = S2T2

    For example. :-

  • If a car covers the distance from A to B in 9 hours with the speed of 45 km/h, then in how much time will it cover this distance with the speed of 36 km/h ?

    Solution. : -
    given : -
    $$S_1=45\,km/h.$$ $$S_2=36\,km/h$$ $$T_1=9\,h.$$ $$T_2= ?$$

    $$45 \times 9=36 \times T_2$$ $$T_2=\frac{45 \times 9}{36}$$ $$T_2=\frac{45}{4}$$ $$T_2=11\frac{1}{4}\,h$$

Type 2 : -

  • If a distance D1 is covered in T1 time and D2 distance in T2 time with a certain speed,then
    $$\frac{D_1}{T_1}=\frac{D_2}{T_2}$$ For example. :-

  • A train covers a distance of 1440 kilometer in 16 hrs with the speed of 60 km/hr. In what time will it cover a distance of 480 kilometer with the same speed ?

    Solution :-
    $$\frac{D_1}{T_1}=\frac{D_2}{T_2}$$ $$here,D_1=1440\,km$$ $$D_2=480\,km$$ $$T_1=16\,h$$ $$T_2= ?$$ $$\frac{1440}{16}=\frac{480}{T_2}$$ $$T_2=\frac{480 \times 16}{1440}$$ $$T_2=5\frac{1}{3}$$ $$Ans.=5\frac{1}{3}\,h$$

Type 3 : -

  • If D1 distance is covered by S1 speed and D2 distance by S2 speed in a given time, then,
    $$\frac{D_1}{S_1}=\frac{D_2}{S_2}$$ example. :-
  • A car covers a distance of 52 km in 4 hours at a speed of 13 km/h. If it increases its speed by 7 km/h in the same time, then how much distance will it cover?

    Solution. : -
    $$\frac{52}{13}=\frac{D_2}{13 + 7}$$ $$\frac{52}{13}=\frac{D_2}{20}$$ $$D_2 =\frac{52 \times 20}{13}=80$$ $$Ans.=80\,km$$

Type 4 : -

  • If two unequal distances d1 and d2 are covered with different speeds x and y respectively, then the average speed $$=\frac{(d_1+d_2) \times x \times y}{d_1 \times y+d_2 \times x}$$ example : -
    • A train covers a distance of 60 km from A to B at a speed of 40 km/hr and from B to C covers a distance of 90 km at a speed of 60 km/hr. What was his average speed during the whole journey?

      Solution : - $$Average\,Speed=\frac{(60+90)\times40\times60}{60\times60+90\times40}$$ $$Average\,Speed=\frac{150\times40\times60}{7200}$$ $$Average\,Speed=50\,km/h$$

Type 5 : -

  • If the distance between two stations is D and two trains run from both the stations towards each other with speed of X and Y, then the time taken by both of them to meet, is $$=\frac{D}{X + Y}$$ example : -
  • If the distance between two stations is 400 km. Two trains move towards each other at 45 km/hr and 35 km/hr from both the stations. After how much time will the two trains meet each other?
    Solution : - $$Total\,Time=\frac{400}{45+35}$$ $$=\frac{400}{80}=5\;hours$$

In case of moving trains,three different situation need to be considered.

  • When a train passes a stationary point ,the distance covered (in the passing) is the length of the train.
  • If the train is crossing a platform or a bridge, the distance covered by train is equal to the length of the train plus length of the plateform or bridge.
  • If two trains pass each other (traveling in the same direction or in opposite directions), the total distance covered (in the crossing or the overtaking, the case may be) is equal to the sum of the lengths of the two trains.
    1. When two trains are moving in opposite direction their speeds should be added to find relative speed.
    2. When they are moving in the same direction the relative speed is the difference of their speeds.

General rules for solving boats and stream problems.

  • When an object is moving in the direction in which the water in the stream is flowing, then the object is said to be moving downstream.
  • When an object is moving against or opposite direction in which the water in the stream is flowing, then the object is said to be moving upstream.
  • When an object is moving in the water where there is no motion in the water, the object can move in any direction with a uniform speed, then the object is said to be moving in still water.
  • Let the speed of the boat in still water = x and speed of the stream is y km/h, then
    1. Speed of the boat with stream = (x + y) km/h.
    2. Speed of the boat against stream = (x - y) km/h.
      As, When the boat is moving downstream, speed of the water add the speed of the boat,
      and when the boat is moving upstream, speed of the water reduces the speed of the boat.
  • If the speed of downstream is $a$ km/h and the speed of upstream is $b$ km/h, then$$Speed\,in\,still\,water=\frac{1}{2}(a + b)\,km/h$$ $$Rate\,of\,stream=\frac{1}{2}(a - b)\,km/h$$

Relative Speed

  • The speed of one (moving) body in the relation to another moving body is called the relative speed of these two bodies, i.e., it is the speed of one moving body as observed, from the second moving body.
  • If two bodies are moving in the same direction, the relative speed is equal to the difference of the speeds of the two bodies.
  • If two bodies are moving in opposite directions, the relative speed is equal to the sum of the speeds of the two bodies.

Example : -

  • A 80 meter long Train passes a plateform of 120 meter long in 10 second.Find the speed of the train.

    Solution : - $$Total \;distance \;covered\; by\; the\; train = length\; of\; the \; train + length\; of\; the\;platform$$

$$Total \;Distance = 80+120=200$$

$$Time = 10 \;seconds$$

$$Speed = \frac{Total \;Distance}{Time}$$

$$Speed=\frac{200}{10}$$

$$Speed=20\; m/s$$

$$In \;km/h$$

$$=20 \times \frac{18}{5}$$

$$Speed=72\; km/h$$

Work, distance and Time Exercise