Linear Motion - Mathematics Form 2 Notes

• Introduction
• Speed
• Velocity and Acceleration
• Distance—time Graph
• Velocity—time Graph
• Relative Speed
• Past KCSE Questions on the Topic.

Introduction

• Distance between the two points is the length of the path joining them while displacement is the distance in a specified direction

Speed

Average speed = distance covered
                              time taken
Example

A man walks for 40 minutes at 60 km/hour, then travels for two hours in a minibus at 80 km/hour. Finally, he travels by bus for one hour at 60 km/h. Find his speed for the whole journey.

Solution

Average speed = distance covered
                             time taken
Total distance =(⁴⁰/₆₀ x 60)km + (2 x 80)km + (1 x 60)km = 260 km
Total time = ⁴/₆ + 2 + 1 = 3²/₃ hrs
Average speed = 260
                         3²/₃
=260 x 3 = 70.9 km/h
       11

Velocity and Acceleration

• For motion under constant acceleration;
  
  Average velocity = initial velocity + final velocity
                                                   2

Example

A car moving in a given direction under constant acceleration. If its velocity at a certain time is 75 km/h and 1 0 seconds later its 90 km/hr.

Solution

Accelaration = change in velocity
                         time taken
=(90 − 75)km/h
        10s
= (90 − 75) x 1000 m/s²
       10 x 60 x 60
= ⁵/₁₂ m/s²

Example

A car moving with a velocity of 50 km/h then the brakes are applied so that it stops after 20 seconds. In this case the final velocity is 0 km/h and initial velocity is 50 km/h.

Solution

Acceleration = (0−50) x 1000 m/s²
                        20 x 60 x 60
= −²⁵/₃₆ m/s²
Negative acceleration is always referred to as deceleration or retardation

Distance Time Graph

• When distance is plotted against time, a distance time graph is obtained.
  
  
• When describing the motion of an object try to be as detailed as possible. For instance...
  • During 'Part A' of the journey the object travels +8m in 4s. It is travelling at a constant velocity of +2ms^-1
  • During 'Part B' of the journey the object travels 0m in 3s. It is stationary for 3 seconds
  • During 'Part C' of the journey the object travels -8m in 3s. It is travelling at a 'constant velocity' of '-2.7ms^-1' back to its starting point, our reference point 0.

Velocity—time Graph

• When velocity is plotted against time, a velocity time graph is obtained.
  
  
• The distance travelled is the area under the graph
• The acceleration and deceleration can be found by finding the gradient of the lines.

Relative Speed

• Consider two bodies moving in the same direction at different speeds. Their relative speed is the difference between the individual speeds.

Example

A van left Nairobi for kakamega at an average speed of 80 km/h. After half an hour, a car left Nairobi for Kakamega at a speed of 100 km/h.

1. Find the relative speed of the two vehicles.
2. How far from Nairobi did the car over take the van

Solution

Relative speed = difference between the speeds
= 100 – 80
= 20 km/h
Distance covered by the van in 30 minutes
Distance = ³⁰/₆₀ x 80 = 40 km
Time taken for car to overtake matatu = ⁴⁰/₂₀
= 2 hours
Distance from Nairobi = 2 x 100 =200 km

Example

A truck left Nyeri at 7.00 am for Nairobi at an average speed of 60 km/h. At 8.00 am a bus left Nairobi for Nyeri at speed of 1 20 km/h .How far from nyeri did the vehicles meet if Nyeri is 1 60 km from Nairobi?

Solution

Distance covered by the lorry in 1 hour = 1 x 60
= 60 km
Distance between the two vehicle at 8.00 am = 160 – 1 00
= 100km
Relative speed = 60 km/h + 120 km/h

Time taken for the vehicle to meet = ¹⁰⁰/₁₈₀
=⁵/₉ hours
Distance from Nyeri = 60 x ⁵/₉ x 60
= 60 + 33.3
= 93.3 km

Past KCSE Questions on the Topic.

1. A bus takes 195 minutes to travel a distance of (2x + 30) km at an average speed of (x - 20) km/h Calculate the actual distance traveled. Give your answers in kilometers.
2. The table shows the height metres of an object thrown vertically upwards varies with the time t seconds. The relationship between s and t is represented by the equations s = at²+bt +10 where b are constants.
   

   t	0	1	2	3	4	5	6	7	8	9	10
   s		45.1

   1.  
      1. Using the information in the table, determine the values of a and b ( 2 marks)
      2. Complete the table (1 mark)
   2.  
      
      1. Draw a graph to represent the relationship between s and t (3 marks)
      2. Using the graph determine the velocity of the object when t = 5 seconds (2 marks)
3. Two Lorries A and B ferry goods between two towns which are 31 20 km apart. Lorry A traveled at km/h faster than lorry B and B takes 4 hours more than lorry A to cover the distance.Calculate the speed of lorry B 
4. A matatus left town A at 7 a.m. and travelled towards a town B at an average speed of 60 km/h. A second matatus left town B at 8 a.m. and travelled towards town A at 60 km/h. If the distance between the two towns is 400 km, find;
   1. The time at which the two matatus met
   2. The distance of the meeting point from town A
5. The figure below is a velocity time graph for a car.
   
   
   1. Find the total distance traveled by the car. (2 marks)
   2. Calculate the deceleration of the car. (2 marks)
6. A bus started from rest and accelerated to a speed of 60 km/h as it passed a billboard. A car moving in the same direction at a speed of 100 km/h passed the billboard 45 minutes later. How far from the billboard did the car catch up with the bus? (3mks)
7. Nairobi and Eldoret are each 250km from Nakuru. At 8.1 5am a lorry leaves Nakuru for Nairobi. At 9.30am a car leaves Eldoret for Nairobi along the same route at 1 00km/h. Both vehicles arrive at Nairobi at the same time.
   1. Calculate their time of arrival in Nairobi (2 marks)
   2. Find the cars speed relative to that of the lorry. (4 marks)
   3. How far apart are the vehicles at 1 2.45pm. (4 marks)
8. Two towns P and Q are 400 km apart. A bus left P for Q. It stopped at Q for one hour and then started the return journey to P. One hour after the departure of the bus from P, a trailer also heading for Q left P. The trailer met the returning bus ¾ of the way from P to Q. They met t hours after the departure of the bus from P.
   1. Express the average speed of the trailer in terms of t
   2. Find the ration of the speed of the bus so that of the trailer.
9. The athletes in an 800 metres race take 104 seconds and 108 seconds respectively to complete the race. Assuming each athlete is running at a constant speed. Calculate the distance between them when the faster athlete is at the finishing line.
10. A and B are towns 360 km apart. An express bus departs form A at 8 am and maintains an average speed of 90 km/h between A and B. Another bus starts from B also at 8 am and moves towards A making four stops at four equally spaced points between B and A. Each stop is of duration 5 minutes and the average speed between any two spots is 60 km/h. Calculate distance between the two buses at 10 am.
11. Two towns A and B are 220 km apart. A bus left town A at 11 . 00 am and traveled towards B at 60 km/h. At the same time, a matatu left town B for town A and traveled at 80 km/h. The matatu stopped for a total of 45 minutes on the way before meeting the bus. Calculate the distance covered by the bus before meeting the matatu.
12. A bus travels from Nairobi to Kakamega and back. The average speed from Nairobi to Kakamega is 80 km/hr while that from Kakamega to Nairobi is 50 km/hr, the fuel consumption is 0.35 litres per kilometer and at 80 km/h, the consumption is 0.3 litres per kilometer .Find
    1. Total fuel consumption for the round trip
    2. Average fuel consumption per hour for the round trip.
13. The distance between towns M and N is 280 km. A car and a lorry travel from M to N. The average speed of the lorry is 20 km/h less than that of the car. The lorry takes 1h 10 min more than the car to travel from M and N.
    1. If the speed of the lorry is x km/h, find x (5 marks)
    2. The lorry left town M at 8: 15 a.m. The car left town M and overtook the lorry at 12.15 p.m.
       Calculate the time the car left town M. 
14. A bus left Mombasa and traveled towards Nairobi at an average speed of 60 km/hr. after 2¹/₂ hours; a car left Mombasa and traveled along the same road at an average speed of 100 km/ hr. If the distance between Mombasa and Nairobi is 500 km, Determine
    1.  
       1. The distance of the bus from Nairobi when the car took off (2mks)
       2. The distance the car traveled to catch up with the bus
    2. Immediately the car caught up with the bus
    3. The car stopped for 25 minutes. Find the new average speed at which the car traveled in order to reach Nairobi at the same time as the bus.
15. A rally car traveled for 2 hours 40 minutes at an average speed of 120 km/h. The car consumes an average of 1 litre of fuel for every 4 kilometers. A litre of the fuel costs Kshs 59. Calculate the amount of money spent on fuel
16. A passenger notices that she had forgotten her bag in a bus 1 2 minutes after the bus had left. To catch up with the bus she immediately took a taxi which traveled at 95 km/hr. The bus maintained an average speed of 75 km/hr. Determine
    
    1. The distance covered by the bus in 12 minutes
    2. The distance covered by the taxi to catch up with the bus
17. The athletes in an 800 metre race take 1 04 seconds and 1 08 seconds respectively to complete the race. Assuming each athlete is running at a constant speed. Calculate the distance between them when the faster athlete is at the finishing line.
18. Mwangi and Otieno live 40 km apart. Mwangi starts from his home at 7.30 am and cycles towards Otieno’s house at 1 6 km/ h Otieno starts from his home at 8.00 and cycles at 8 km/h towards Mwangi at what time do they meet?
19. A train moving at an average speed of 72 km/h takes 1 5 seconds to completely cross a bridge that is 80m long.
    1. Express 72 km/h in metres per second
    2. Find the length of the train in metres