# Time, Speed and Distance

Pratice Time, Speed and Distance Questions and answers.

Rule Description
1

Average speed =

$\frac{\text{Total Distance}}{\text{Total time}}$

2

While traveling a certain distance d, if a man changes his speed in the ratio m : n, then the ratio of time taken becomes n : m.

3

If a certain distance d, from A to B, is covered at a km/hr and the same distance is covered again from B to A in b km/hr, then the average speed during the whole journey is given by:

Average speed =

$\frac{2ab}{a+b}$ km/hr.

4

If t1 and t2 is time taken to travel from A to B and B to A respectively, the distance d from A to B is given by :

$d=(t_1 -t_2 )\left(\frac{ab}{a+b}\right)$

5

If first part of the distance is covered at the rate of v1 in time t1 and the second part of the distance is covered at the rate of v2 in time t2, then the average speed is

$\left(\frac{v_1t_1+v_2t_2}{t_1+t_2}\right)$

### Relative Speed

Rule Description
1
• When two bodies are moving in opposite direction with speeds S1 and S2 respectively, their relative speed =

$S_2 + S_1$

• When two bodies are moving in same direction with speeds S1 and S2 respectively, their relative speed =

$S_2 - S_1$

2

If two persons start at the same time in opposite directions from two points A and B, and after crossing each other they take x and y hours respectively to complete the journey, then

$\frac{\text{Speed of first}}{\text{Speed of second}}=\sqrt{\frac{y}{x}}$

3

If a man changes his speed to $\frac{a}{b}$ of his usual speed, reaches his destination late/earlier by t minutes then,

usual time = $\frac{t}{\left(\frac{b}{a}-1\right)}$

4

A man covers a certain distance D. If he moves S1 speed faster, he would have taken t time less and if he moves S2 speed slower, he would have taken t time more. The original speed is given by:

$\frac{2\left(S_1\times S_2\right)}{S_2-S_1}$

5

If a person with two different speeds U & V cover the same distance, then required distance:

• $\frac{U\times V}{U-V} \times \text{Difference between arrival time}$
• (OR)

• $\frac{U\times V}{U+V} \times \text{Total time taken}$
6

A policemen sees a thief at a distance of d. He starts chasing the thief who is running at a speed of ‘a’ and policeman is chasing with a speed of ‘b’. In this case, the distance covered by the thief when he is caught by the policeman, is given by:

$d\left(\frac{a}{b-a}\right)$

7

A man leaves a point A at t1 and reaches the point B at t2. Another man leaves the point B at t3 and reaches the point A at t4, then they will meet at:

$t_1+\frac{\left(t_2-t_1\right)\left(t_4-t_1\right)}{\left(t_2-t_1\right)\left(t_4-t_3\right)}$

8

Relation between time taken with two different modes of transport:

$t_{2x}+t_{2y}=2\left(t_x+t_y\right)$

where,

• tx = time when mode of transport x is used single way.
• ty = time when mode of transport y is used single way.
• t2x = time when mode of transport x is used both ways.
• t2y = time when mode of transport y is used both ways.

### Train

Rule Description
1

Time taken by a train to cross a pole =

$\frac{\text{Length of train}}{\text{Speed of train}}$

2

Time taken by a train to cross platform =

$\frac{\text{length of train + length of platform}}{\text{Speed of train}}$

3

When two trains with lengths L1 and L2 and with speeds S1 and S2 respectively, then

(a) When they are moving in the same direction, time taken by the faster train to cross the slower train =

$\frac{L_1 + L_2}{\text{Difference of their speeds}}$

4

(b) When they are moving in the opposite direction, time taken by the faster train to cross each other =

$\frac{L_1 + L_2}{\text{Sum of their speeds}}$

5

Suppose two trains of lengths x km and y km are moving in the same direction at u km/hr and v km/hr respectively, then

• Their relative speed = $(u – v)$ km/hr.
• Time taken by the faster train to cross the slower train =
$\left(\frac{x+y}{u-v}\right)$ km/hr.
6

Suppose two trains of lengths x km and y km are moving in opposite direction at u km/hr and v km/hr respectively, then

• Their relative speed = $(u + v)$ km/hr.
• Time taken by the faster train to cross the slower train =
$\left(\frac{x+y}{u+v}\right)$ km/hr.
7

If a man is running at a speed of u m/sec in the same direction in which a train of length L meters is running at a speed v m/sec, then

• Speed of the train relative to man = $(v – u)$ m/sec.
• Time taken by the train to cross the man = $\frac{1}{v-u}$ seconds.
8

If a man is running at a speed of u m/sec in a direction opposite to that in which a train of length L meters is running at a speed v m/sec, then

• Speed of the train relative to man = $(v + u)$ m/sec.
• Time taken by the train to cross the man = $\frac{1}{v+u}$ seconds.
9

If two trains start at the same time from two points A and B towards each other and after crossing, they take (a) and (b) hours in reaching B and A respectively. Then,

A’s speed : B’s speed = $\left(\sqrt{b}:\sqrt{a}\right)$

10

If a train of length L m passes a platform of x m in t1 seconds, then time taken t2 by the same train to pass a platform of length y m is given as

$t_2=\left(\frac{L+y}{L+x}\right)t_1$

11

From stations P and Q, two trains start moving towards each other with the speeds a and b, respectively. When they meet each other, it is found that one train covers distance d more than that of another train. In such cases,

distance between stations P and Q is given as

$\left(\frac{a+b}{a-b}\right)\times d$

12

The distance between two stations P and Q is d km. A train with speed a km/h starts from station P towards Q and after a difference of t hr another train with b km/h starts from Q towards station P, then both the trains will meet at a certain point after time T. Then,

• If first train starts before the first train
$T=\left(\frac{d + tb}{a+b}\right)$
• If second train starts before the first train
$T=\left(\frac{d - tb}{a+b}\right)$
13

The distance between two stations P and Q is d km. A train starts from P towards Q and another train starts from Q towards P at the same time and they meet at a certain point after t h. If the difference between the two trains is x km/h, then

• Speed of faster train = $\left(\frac{d+tx}{2t}\right)$ km/h
• Speed of slower train = $\left(\frac{d-tx}{2t}\right)$ km/h
14

A train covers distance d between two stations P and Q in t1 h. If the speed of train is reduced by a km/h, then the same distance will be covered in t2 h.

• Distance between P and Q is
$d=a\left(\frac{t_1t_2}{t_2-t_1}\right)$ km
• Speed of the train =
$\left(\frac{at_2}{t_2-t_1}\right)$ km/h