Newton's Law of Motion | Jamb(UTME)

Jamb(utme) key points on Newtons' laws; inertia; mass; force; relationship between mass and accelaration; impulse and momentum

Newton's Laws of Motion

1. First Law (Law of Inertia): An object remains at rest or in uniform motion in a straight line unless acted upon by an external force.
2. The First Law explains why objects don’t move unless a force is applied (e.g., a stationary chair stays at rest until pushed).
3. Second Law (Law of Acceleration): The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass:
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   $F = ma$
4. The Second Law quantifies the relationship between force, mass, and acceleration.
5. Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.
6. The Third Law explains phenomena like recoil in guns or the thrust in rockets.
7. Newton’s Laws are the foundation of classical mechanics.
8. These laws apply to objects in motion or at rest as long as relativistic speeds or quantum effects are negligible.
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Inertia

9. Inertia is the property of matter that resists changes in motion or rest.
10. Greater mass means greater inertia.
11. A stationary truck is harder to move than a bicycle due to higher inertia.
12. Inertia applies to rotational motion (rotational inertia depends on the mass distribution around an axis).
13. Newton's First Law is also known as the Law of Inertia.
14. Inertia explains why passengers lurch forward in a sudden stop or backward during acceleration.
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Mass

15. Mass is the measure of the amount of matter in an object.
16. The SI unit of mass is the kilogram $kg$.
17. Mass is scalar and does not change with location (unlike weight).
18. Mass is a measure of an object's resistance to acceleration.
19. Weight is the force due to gravity acting on mass:
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    $W = mg$
20. Mass is proportional to inertia; larger masses are harder to accelerate.
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Force

21. Force is an interaction that changes the motion of an object.
22. The SI unit of force is the newton $N$.
23. Force is a vector quantity with both magnitude and direction.
24. A force can accelerate, decelerate, or change the direction of an object.
25. The net force is the vector sum of all forces acting on an object.
26. Balanced forces result in no acceleration, while unbalanced forces cause motion.
27. Common types of forces include gravitational, frictional, and applied forces.
28. The formula for force is:
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    $F = ma$
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Relationship Between Mass and Acceleration

29. Acceleration is inversely proportional to mass for a constant force:
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    $a = \frac{F}{m}$
30. Objects with greater mass require more force to achieve the same acceleration as lighter objects.
31. For example, pushing a small car is easier than pushing a truck due to the difference in mass.
32. Mass affects acceleration, but force determines the magnitude of motion.
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Impulse

33. Impulse is the product of force and the time over which it acts: $J = F \Delta t$
34. The SI unit of impulse is $N·s$ (newton-second).
35. Impulse is a vector quantity.
36. Impulse equals the change in momentum:
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    $J = \Delta p$
37. Longer impact times result in smaller forces (e.g., airbags).
38. Shorter impact times create higher forces (e.g., a hammer hitting a nail).
39. Impulse explains why following through in sports increases momentum.
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Momentum

40. Momentum is the product of an object's mass and velocity:
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    $p = mv$
41. Momentum is a vector quantity with the same direction as velocity.
42. The SI unit of momentum is $kg·m/s$.
43. An object at rest has zero momentum.
44. Momentum is conserved in isolated systems (Law of Conservation of Momentum).
45. In collisions, total momentum before impact equals total momentum after, assuming no external forces.
46. Elastic collisions conserve both momentum and kinetic energy.
47. Inelastic collisions conserve momentum but not kinetic energy.
48. Momentum depends equally on mass and velocity; doubling either doubles the momentum.
49. Momentum conservation explains rocket propulsion and the recoil of guns.
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Impulse-Momentum Theorem

50. Impulse causes a change in momentum: $J = \Delta p$
51. This theorem is used to calculate the effects of forces in collisions and impacts.
52. For example, the momentum change in a bouncing ball depends on the impulse exerted by the surface.
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Applications in Everyday Life

53. Seatbelts reduce force by increasing the time over which momentum changes during car crashes.
54. In sports, follow-through helps impart greater momentum to a ball.
55. Airbags extend impact time, reducing the force on passengers.
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Engineering

56. Crumple zones in cars reduce momentum transfer to passengers during crashes.
57. Rocket engines generate thrust by expelling gases to conserve momentum.
58. Turbines use momentum changes in fluids to generate mechanical work.
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Physics and Astronomy

59. In space, astronauts move by pushing off surfaces, demonstrating momentum conservation.
60. In particle physics, momentum conservation helps analyze collisions in particle accelerators.
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Jamb(utme) key points on force-time graph; conservation of linear momentum and application

Force-Time Graph**

1. A force-time graph shows how the force acting on an object varies over time.
2. The x-axis represents time $(t)$ and the y-axis represents force $(F)$.
3. The area under a force-time graph represents impulse.
4. Impulse is the product of force and the time over which it acts.
5. The formula for impulse is:
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   $J = \int F(t) \, dt$
6. Impulse is equal to the change in momentum of the object: $J = \Delta p$
7. If the force is constant, the area under the graph is a rectangle $(J = F \cdot \Delta t)$.
8. If the force varies, the area can be calculated using geometry (triangles, trapezoids) or integration.
9. A spike in the graph indicates a sudden large force acting over a very short time.
10. The slope of the force-time graph is not directly meaningful but indicates how quickly the force changes.
11. In collisions, force-time graphs often show a sharp rise and fall, representing impact forces.
12. Longer impact times (wider graph) reduce peak forces, minimizing damage (e.g., airbags in cars).
13. A flat line on the graph indicates a constant force.
14. A zero line indicates no force acting on the object.
15. Negative areas below the x-axis represent forces acting in the opposite direction.
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Conservation of Linear Momentum

16. Momentum $(p)$ is the product of mass and velocity: $p = mv$
17. The Law of Conservation of Linear Momentum states that in an isolated system (no external forces), the total momentum remains constant.
18. Mathematically: $p_{initial} = p_{final}$
19. The principle applies to collisions, explosions, and any interaction between objects.
20. Internal forces, like those during a collision, do not affect the system's total momentum.
21. Momentum conservation is valid for both elastic and inelastic collisions.
22. In elastic collisions, both momentum and kinetic energy are conserved.
23. In inelastic collisions, only momentum is conserved, while kinetic energy is not.
24. Momentum conservation also applies in two-dimensional motion, where components in each direction are conserved separately.
25. In explosions, the total momentum before and after the event remains zero if the system was initially at rest.
26. Momentum conservation explains how objects recoil after releasing energy (e.g., firing a gun).
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Applications of Conservation of Momentum

Recoil and Explosions

31. A gun recoils when fired due to conservation of momentum.
32. Rockets work based on momentum conservation: expelling gases downward creates upward thrust.
33. In fireworks, the fragments of an explosion spread out in different directions while conserving total momentum.
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Astronomy and Space

34. Satellites use momentum exchange during docking and separation from spacecraft.
35. In space, astronauts move by pushing off objects, relying on momentum conservation.
36. Collisions between celestial bodies, like asteroids, are analyzed using momentum conservation.
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Vehicle Design

37. Momentum conservation principles are applied in designing crumple zones to reduce crash impact forces.
38. Airbags increase the time over which momentum is transferred, reducing peak forces on passengers.
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Engineering and Physic

39. Momentum conservation is used to design systems like jet engines and turbines.
40. The study of particle collisions in physics relies on momentum conservation to understand subatomic interactions.
41. Pendulums and Newton’s cradle demonstrate momentum conservation in motion.
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Everyday Scenarios

42. When a person jumps out of a stationary boat, the boat moves backward due to momentum conservation.
43. Momentum transfer allows a skateboarder to propel forward by pushing against the ground.
44. A child sliding on ice throws an object backward to move forward.
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Industrial Applications

45. Hydraulic presses and hammers use momentum principles to deliver large forces during impact.
46. Conveyor belts rely on momentum transfer to move goods efficiently.
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Traffic Safety

47. Analyzing traffic accidents involves calculating the pre- and post-collision momentum to reconstruct events.
48. Seatbelts help spread the force over time, reducing the momentum change experienced by passengers.
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Sports

49. Catchers in baseball reduce force by pulling their hands back, increasing the time of momentum transfer.
50. In soccer, players impart momentum to the ball by kicking it with varying force and direction.
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Jamb(utme) Calculation problems and solutions involving impulse and momentum

1. Calculating Momentum

2. Momentum After Velocity Change

3. Impulse from Force and Time

4. Impulse from Momentum Change

5. Force from Impulse

6. Velocity After Impulse

7. Impulse with Direction Change

8. Force Over Time

9. Time for a Given Force

10. Velocity from Momentum

11. Final Momentum

12. Momentum Conservation

13. Final Velocity (Elastic Collision)

14. Force Acting Over Time

15. Finding Impulse

16. Momentum After Collision

• Total momentum before collision: $p_{initial} = (6 \cdot 4) + (3 \cdot 0) = 24kg·m/s$
• Total mass after collision: $m_{\text{total}} = 6 + 3 = 9kg$
• Final velocity: $v_{final} = \frac{p_{initial}}{m_{total}} = \frac{24}{9} = 2.67m/s$
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17. Momentum in Two Dimensions

• Momentum components: $p_x = 2 \cdot 3 = 6kg·m/s, \quad p_y = 4 \cdot 2 = 8kg·m/s$
• Total momentum magnitude: $p_{total} = \sqrt{p_x^2 + p_y^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10kg·m/s$
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18. Impulse to Stop

19. Time to Stop an Object

• Impulse required: $J = \Delta p = m \cdot v = 5 \cdot 20 = 100N·s$
• Time: $t = \frac{J}{F} = \frac{100}{25} = 4s$
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20. Force During Collision

• Impulse: $J = \Delta p = m \cdot v = 0.5 \cdot 10 = 5N·s$
• Force: $F = \frac{J}{t} = \frac{5}{0.2} = 25N$
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21. Velocity After Force Application

• Impulse: $J = F \cdot t = 9 \cdot 5 = 45N·s$ $
• Final velocity: $v_{final} = \frac{J}{m} = \frac{45}{3} = 15m/s$ $
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22. Elastic Collision

• Velocity formula for elastic collision: $v_1' = \frac{(m_1 - m_2)v_1 + 2m_2v_2}{m_1 + m_2}$ Substituting values: $v_1' = \frac{(1 - 2)4 + 2 \cdot 2 \cdot 0}{1 + 2} = \frac{-4}{3} = -1.33m/s$
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23. Recoil Velocity

• Momentum conservation: $m_1v_1 + m_2v_2 = 0$ Solving for $v_2:$ $v_2 = -\frac{m_1v_1}{m_2} = -\frac{50 \cdot 3}{200} = -0.75m/s$
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24. Velocity of a System After Explosion

• Momentum conservation: $6 \cdot 0 = 4 \cdot 5 + 2 \cdot v$ Solving for $v$: $v = -\frac{4 \cdot 5}{2} = -10 \, \text{m/s}$
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25. Kinetic Energy Loss in Inelastic Collision

• Initial kinetic energy: $KE_{initial} = \frac{1}{2} m v^2 = \frac{1}{2} \cdot 2 \cdot 6^2 = 36J$
• Final velocity: $v_{final} = \frac{p_{total}}{m_{total}} = \frac{2 \cdot 6}{2 + 4} = 2m/s$
• Final kinetic energy: $KE_{final} = \frac{1}{2} m_{total} v_{final}^2 = \frac{1}{2} \cdot 6 \cdot 2^2 = 12J$
• Energy lost: $KE_{lost} = 36 - 12 = 24J$
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26. Impulse in a Bouncing Ball

27. Impulse During a Collision

28. Explosion Momentum Conservation

• Momentum conservation: $10 \cdot 0 = (6 \cdot 2) + (4 \cdot v)$ Solving for ( v ): $v = -\frac{6 \cdot 2}{4} = -3m/s$
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29. Impulse from Average Force

30. Final Velocity After Recoil

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