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mpt · CSS study notes

Time, Speed & Distance

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Quick recall

  • Average speed = total distance divided by total time.
  • Use the equal-distance or equal-time shortcut only when its condition holds.

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Formula Cards

1. Distance, speed, time

D=ST,S=DT,T=DSD=ST,\qquad S=\frac{D}{T},\qquad T=\frac{D}{S}

2. Convert km/h to m/s

vm/s=vkm/h518v_{\text{m/s}}=v_{\text{km/h}}\frac{5}{18}

3. Convert m/s to km/h

vkm/h=vm/s185v_{\text{km/h}}=v_{\text{m/s}}\frac{18}{5}

4. Average speed — always valid

vˉ=Total distanceTotal time\bar{v}=\frac{\text{Total distance}}{\text{Total time}}

5. Equal distances

vˉ=2v1v2v1+v2\bar{v}=\frac{2v_1v_2}{v_1+v_2}

6. Equal times

vˉ=v1+v22\bar{v}=\frac{v_1+v_2}{2}

7. Relative speed, same direction

vrelative=∣v1−v2∣v_{\text{relative}}=|v_1-v_2|

8. Relative speed, opposite directions

vrelative=v1+v2v_{\text{relative}}=v_1+v_2

Worked Examples

Example 1

Question: Travel 60 km at 30 km/h and return 60 km at 60 km/h. Find average speed.

Step 1: Find total distance and time.

D=120 km,T=6030+6060=3 hD=120\text{ km},\qquad T=\frac{60}{30}+\frac{60}{60}=3\text{ h}

Step 2: Divide distance by time.

vˉ=1203=40 km/h\bar{v}=\frac{120}{3}=40\text{ km/h}

Answer: 40 km/h.

Example 2

Question: Convert 72 km/h to m/s.

Step 1: Use the conversion factor.

v=72×518v=72\times\frac{5}{18}

Step 2: Calculate.

v=20 m/sv=20\text{ m/s}

Answer: 20 m/s.

Example 3

Question: Two cars 300 km apart approach at 40 and 60 km/h. When do they meet?

Step 1: Add approaching speeds.

vrelative=40+60=100 km/hv_{\text{relative}}=40+60=100\text{ km/h}

Step 2: Divide separation by relative speed.

T=300100=3 hT=\frac{300}{100}=3\text{ h}

Answer: 3 hours.

Example 4

Question: A travels for half the total journey time at 40 km/h and half at 60 km/h. Find average speed.

Step 1: Recognize that the time intervals are equal.

T1=T2T_1=T_2

Step 2: Use the equal-time average.

vˉ=40+602=50 km/h\bar{v}=\frac{40+60}{2}=50\text{ km/h}

Answer: 50 km/h. If the two distances were equal instead, the answer would be 48 km/h.

Drill

Try each question before opening its answer.

Question 1

Convert 90 km/h to m/s.

  • Show answer

    25 m/s

Question 2

Convert 15 m/s to km/h.

  • Show answer

    54 km/h

Question 3

A goes 100km at 50km/h, returns at 25km/h. Avg speed?

  • Show answer

    33.33 km/h

Question 4

Two trains 500km apart approach at 50 and 75 km/h. Time to meet?

  • Show answer

    4h

Question 5

A car travels at 60km/h for 2h then 40km/h for 3h. Average speed for whole trip?

  • Show answer

    48 km/h

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