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Lesson: Newton's Second Law of Motion in Vector Notation

Sample Question Videos

Worksheet • 19 Questions • 1 Video

Q1:

If a body of mass 1 kg moves under the action of forces ⃑ 𝐹 = ο€» ⃑ 𝑖 + 8 ⃑ 𝑗 βˆ’ 5 ⃑ π‘˜  1 N and ⃑ 𝐹 = ο€» 2 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗 + 8 ⃑ π‘˜  2 N , what is its acceleration?

  • A ο€» 3 ⃑ 𝑖 + ⃑ 𝑗 + 3 ⃑ π‘˜  m/s2
  • B ο€» 6 ⃑ 𝑖 + ⃑ 𝑗 + 3 ⃑ π‘˜  m/s2
  • C ο€» 3 ⃑ 𝑖 + ⃑ 𝑗 + 9 ⃑ π‘˜  m/s2
  • D ο€» 3 ⃑ 𝑖 + 2 ⃑ 𝑗 + 3 ⃑ π‘˜  m/s2

Q2:

A body of mass 11 kg is moving such that the horizontal and vertical components of its velocity are given by 𝑣 = 4 π‘₯ and 𝑣 = βˆ’ 9 . 8 𝑑 + 1 2 𝑦 where 𝑣 π‘₯ and 𝑣 𝑦 are measured in metres per second. Find the force ⃑ 𝐹 , in newtons, that is acting on the body during its motion and the body’s initial speed 𝑣 0 .

  • A 𝑣 = 4 √ 1 0 / 0 m s , ⃑ 𝐹 = βˆ’ 1 0 7 . 8 ⃑ 𝑗
  • B 𝑣 = 4 / 0 m s , ⃑ 𝐹 = 4 ⃑ 𝑖 βˆ’ 1 0 7 . 8 ⃑ 𝑗
  • C 𝑣 = 4 / 0 m s , ⃑ 𝐹 = 4 ⃑ 𝑖 + 2 4 . 2 ⃑ 𝑗
  • D 𝑣 = 4 √ 1 0 / 0 m s , ⃑ 𝐹 = βˆ’ 2 3 9 . 8 ⃑ 𝑗

Q3:

A body of mass 3 units was moving under the action of two coplanar forces ⃑ 𝐹 1 and ⃑ 𝐹 2 such that ⃑ 𝐹 = π‘Ž ⃑ 𝑖 + 4 ⃑ 𝑗 1 and ⃑ 𝐹 = βˆ’ 4 ⃑ 𝑖 + 𝑏 ⃑ 𝑗 2 , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given that the acceleration of the body is 2 ⃑ 𝑖 βˆ’ 4 ⃑ 𝑗 , find the values of the constants π‘Ž and 𝑏 .

  • A π‘Ž = 1 0 , 𝑏 = βˆ’ 1 6
  • B π‘Ž = 2 , 𝑏 = βˆ’ 8
  • C π‘Ž = βˆ’ 2 , 𝑏 = 0
  • D π‘Ž = 6 , 𝑏 = βˆ’ 8

Q4:

A particle of mass π‘š kg is moving under the action of two forces: ⃑ 𝐹 = 8 π‘š ⃑ 𝑖 + 6 π‘š ⃑ 𝑗 1 and ⃑ 𝐹 = 4 π‘š ⃑ 𝑖 2 , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Find the acceleration ⃑ π‘Ž of the particle and its magnitude β€– β€– ⃑ π‘Ž β€– β€– in metres per second squared.

  • A ⃑ π‘Ž = 1 2 ⃑ 𝑖 + 6 ⃑ 𝑗 , β€– β€– ⃑ π‘Ž β€– β€– = 6 √ 5 / m s 2
  • B ⃑ π‘Ž = 4 ⃑ 𝑖 + 6 ⃑ 𝑗 , β€– β€– ⃑ π‘Ž β€– β€– = 2 √ 1 3 / m s 2
  • C ⃑ π‘Ž = 1 2 ⃑ 𝑖 + 6 ⃑ 𝑗 , β€– β€– ⃑ π‘Ž β€– β€– = 6 √ 3 / m s 2
  • D ⃑ π‘Ž = 1 2 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 , β€– β€– ⃑ π‘Ž β€– β€– = 6 √ 5 / m s 2

Q5:

If the forces ⃑ 𝐹 = ο€» π‘₯ ⃑ 𝑖 + 𝑦 ⃑ 𝑗 + 𝑧 ⃑ π‘˜  1 N and ⃑ 𝐹 = ο€» βˆ’ 5 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 βˆ’ 3 ⃑ π‘˜  2 N acting on a body of mass 6 kg, cause an acceleration ⃑ π‘Ž = ο€» 5 ⃑ 𝑖 + 2 ⃑ 𝑗 βˆ’ 4 ⃑ π‘˜  / m s 2 , what are the values of π‘₯ , 𝑦 , and 𝑧 ?

  • A π‘₯ = 3 5 , 𝑦 = 1 8 , 𝑧 = βˆ’ 2 1
  • B π‘₯ = 2 5 , 𝑦 = 6 , 𝑧 = βˆ’ 2 7
  • C π‘₯ = 0 , 𝑦 = βˆ’ 4 , 𝑧 = βˆ’ 7
  • D π‘₯ = 3 0 , 𝑦 = 1 2 , 𝑧 = βˆ’ 2 4

Q6:

Given that the motion of a body of mass 2 kg is represented by the relation ⃑ π‘Ÿ ( 𝑑 ) = ο€Ή 6 𝑑 + 1 5 𝑑 + 2  ⃑ 𝑐 2 , where ⃑ 𝑐 is a constant unit vector, ⃑ π‘Ÿ is measured in metres, and 𝑑 is measured in seconds, determine the magnitude of the force acting on the body.

Q7:

A body of unit mass was moving under the effect of a force ⃑ 𝐹 = π‘Ž ⃑ 𝑖 + 𝑏 ⃑ 𝑗 , where ⃑ 𝑖 and ⃑ 𝑗 are two orthogonal unit vectors. If the displacement vector of the body at time 𝑑 is given by ⃑ 𝑠 ( 𝑑 ) = ( 9 𝑑 ) ⃑ 𝑖 + ( 𝑑 + 3 ) ⃑ 𝑗 2 2 , find π‘Ž and 𝑏 .

  • A π‘Ž = 1 8 , 𝑏 = 2
  • B π‘Ž = 9 , 𝑏 = 1
  • C π‘Ž = 2 , 𝑏 = 1 8
  • D π‘Ž = 1 8 , 𝑏 = 1
  • E π‘Ž = 9 , 𝑏 = 2

Q8:

A particle of unit mass was moving under the effect of three forces: ⃑ 𝐹 = π‘Ž ⃑ 𝑗 1 , ⃑ 𝐹 = βˆ’ ⃑ 𝑖 2 , and ⃑ 𝐹 = 2 ⃑ 𝑗 + 𝑏 ⃑ 𝑖 3 , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors and π‘Ž and 𝑏 are constants. If the displacement vector of the particle as a function of the time is given by ⃑ 𝑠 ( 𝑑 ) = 6 ⃑ 𝑖 + ( βˆ’ 4 𝑑 + 4 𝑑 ) ⃑ 𝑗 2 , find the values of π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 0 , 𝑏 = 1
  • B π‘Ž = 1 0 , 𝑏 = 1
  • C π‘Ž = 1 0 , 𝑏 = βˆ’ 1
  • D π‘Ž = βˆ’ 1 0 , 𝑏 = βˆ’ 1

Q9:

A body of mass 9 g was moving on a plane under the effect of the force ⃑ 𝐹 = ο€Ί βˆ’ ⃑ 𝑖 βˆ’ 1 0 ⃑ 𝑗  dynes. Given that the position vector of the body is given by the relation ⃑ π‘Ÿ ( 𝑑 ) = ο€Ί ο€Ή π‘Ž 𝑑 + 7  ⃑ 𝑖 + ο€Ή 𝑏 𝑑 + 6 𝑑  ⃑ 𝑗  2 2 c m , determine π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 1 8 , 𝑏 = βˆ’ 5 9
  • B π‘Ž = βˆ’ 1 1 8 , 𝑏 = βˆ’ 3 2 9
  • C π‘Ž = βˆ’ 3 2 9 , 𝑏 = βˆ’ 5 9
  • D π‘Ž = βˆ’ 1 2 , 𝑏 = βˆ’ 5

Q10:

A body of mass 7 kg moves under the action of three forces, ⃑ 𝐹 = ο€Ί π‘Ž ⃑ 𝑖 + 3 ⃑ 𝑗  1 N , ⃑ 𝐹 = ο€Ί 6 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗  2 N , and ⃑ 𝐹 = ο€Ί 6 ⃑ 𝑖 + 𝑏 ⃑ 𝑗  3 N . Given that the displacement of the particle at time 𝑑 seconds is ⃑ 𝑠 =  ο€Ή 𝑑 + 6  ⃑ 𝑖 + ο€Ή 5 𝑑 + 5  ⃑ 𝑗  2 2 m , determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = 2 , 𝑏 = 7 3
  • B π‘Ž = βˆ’ 1 0 , 𝑏 = 1 3
  • C π‘Ž = 1 4 , 𝑏 = 6 1
  • D π‘Ž = 1 4 , 𝑏 = 7 9

Q11:

A particle of unit mass is moving such that its velocity at a given time 𝑑 is represented by ⃑ 𝑣 ( 𝑑 ) = ο€Ή 8 π‘Ž 𝑑 + 5 𝑏 𝑑  ⃑ 𝑖 2 , where ⃑ 𝑖 is a constant unit vector. Given that the force acting on the particle at time 𝑑 is ⃑ 𝐹 ( 𝑑 ) = ( 1 0 𝑑 + 4 ) ⃑ 𝑖 , find π‘Ž and 𝑏 .

  • A π‘Ž = 5 8 , 𝑏 = 4 5
  • B π‘Ž = βˆ’ 5 8 , 𝑏 = 4 5
  • C π‘Ž = βˆ’ 5 8 , 𝑏 = βˆ’ 4 5
  • D π‘Ž = 5 8 , 𝑏 = βˆ’ 4 5

Q12:

A particle of unit mass is moving along a certain path, where its velocity at time 𝑑 is given by the relation ⃑ 𝑣 = ο€Ή π‘Ž 𝑑 + 𝑏 𝑑  ⃑ 𝑖 2 , where ⃑ 𝑖 is a constant unit vector. Given that the force acting on the particle is constant and given by the relation ⃑ 𝐹 = 9 1 ⃑ 𝑖 , determine the values of the constants π‘Ž and 𝑏 .

  • A π‘Ž = 0 , 𝑏 = 9 1
  • B π‘Ž = 9 1 , 𝑏 = 0
  • C π‘Ž = βˆ’ 9 1 , 𝑏 = 0
  • D π‘Ž = 0 , 𝑏 = βˆ’ 9 1

Q13:

A body of mass 250 g moves under the action of a force, ⃑ 𝐹 newtons. Given that the body starts from rest at the origin, and ⃑ 𝐹 = ( 9 𝑑 + 3 ) ⃑ 𝑖 + 9 𝑑 ⃑ 𝑗 , where ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors, find the displacement in terms of 𝑑 .

  • A ο€Ή 6 𝑑 + 6 𝑑  ⃑ 𝑖 + ο€Ή 6 𝑑  ⃑ 𝑗 3 2 3
  • B ο€Ή 1 2 𝑑 + 6 𝑑  ⃑ 𝑖 + ο€Ή 6 𝑑  ⃑ 𝑗 3 2 3
  • C ο€Ή 6 𝑑 + 6 𝑑  ⃑ 𝑖 + ο€Ή 1 8 𝑑  ⃑ 𝑗 3 2 3
  • D ο€Ή 6 𝑑 + 1 2 𝑑  ⃑ 𝑖 + ο€Ή 6 𝑑  ⃑ 𝑗 3 2 3

Q14:

A particle of mass 5 kg was in motion. The components of its velocity in the horizontal and vertical directions were 𝑣 = 3 / π‘₯ m s and 𝑣 = ( βˆ’ 4 . 7 𝑑 + 1 4 ) / 𝑦 m s , respectively. Determine the magnitude, 𝑣 0 , and direction, πœƒ , of its initial velocity and the force ⃑ 𝐹 acting on it.

  • A 𝑣 = √ 2 0 5 / 0 m s , πœƒ = 7 7 5 4 β€² ∘ , ⃑ 𝐹 = βˆ’ 2 3 . 5 ⃑ 𝑗
  • B 𝑣 = √ 1 9 9 / 0 m s , πœƒ = 7 2 7 β€² ∘ , ⃑ 𝐹 = 2 3 . 5 ⃑ 𝑗
  • C 𝑣 = √ 2 0 5 / 0 m s , πœƒ = 7 7 5 4 β€² ∘ , ⃑ 𝐹 = βˆ’ 4 . 7 ⃑ 𝑗
  • D 𝑣 = √ 2 3 / 0 m s , πœƒ = 7 2 7 β€² ∘ , ⃑ 𝐹 = βˆ’ 4 . 7 ⃑ 𝑗

Q15:

A body of mass π‘š is moving under the action of a force F . Its velocity at time 𝑑 seconds is given by the relation v i ( 𝑑 ) = ( 6 π‘Ž 𝑑 + 𝑏 ) / m s , where i is a unit vector in the direction of its motion, and π‘Ž and 𝑏 are constants. Given that the initial velocity of the body v i  = 1 5 / m s and F i = ( 1 2 π‘š ) N , find the body’s speed at 𝑑 = 1 4 s e c o n d s .

Q16:

Three forces, ⃑ 𝐹 = ο€» π‘Ž ⃑ 𝑖 + 4 ⃑ 𝑗 βˆ’ 9 ⃑ π‘˜  1 N , ⃑ 𝐹 = ο€» 3 ⃑ 𝑖 βˆ’ 8 ⃑ 𝑗 + 𝑐 ⃑ π‘˜  2 N , and ⃑ 𝐹 = ο€» 4 ⃑ 𝑖 + 𝑏 ⃑ 𝑗 + 8 ⃑ π‘˜  3 N , where ⃑ 𝑖 , ⃑ 𝑗 , and ⃑ π‘˜ are three perpendicular unit vectors, are acting upon a body of unit mass. If the displacement vector of the particle is ⃑ 𝑠 =  ( 4 𝑑 ) ⃑ 𝑖 + ο€Ή 6 𝑑 + 3 𝑑  ⃑ 𝑗 + ο€Ή 8 𝑑 + 7  ⃑ π‘˜  2 2 m , determine the constants π‘Ž , 𝑏 , and 𝑐 .

  • A π‘Ž = βˆ’ 7 , 𝑏 = 1 6 , and 𝑐 = 1 7
  • B π‘Ž = βˆ’ 1 , 𝑏 = 1 0 , and 𝑐 = 9
  • C π‘Ž = βˆ’ 7 , 𝑏 = 1 6 , and 𝑐 = βˆ’ 1
  • D π‘Ž = βˆ’ 1 , 𝑏 = 1 6 , and 𝑐 = 1 7
  • E π‘Ž = βˆ’ 7 , 𝑏 = 1 8 , and 𝑐 = 1 7

Q17:

A body of mass 478 g has an acceleration of ο€Ί βˆ’ 4 ⃑ 𝑖 + 3 ⃑ 𝑗  m/s2, where ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors. What is the magnitude of the force acting on the body?

Q18:

A body of mass 1 kg was moving in a straight line with a velocity ⃑ 𝑣 = ο€Ί 8 ⃑ 𝑖 βˆ’ 8 ⃑ 𝑗  / m s , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. The force ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗  N acted on the body for 8 seconds. Find the body’s speed after the action of this force.

  • A 2 4 √ 5 m/s
  • B 8 √ 6 1 m/s
  • C 40 m/s
  • D 8 √ 4 1 m/s

Q19:

A body of mass 2 kg moves in a horizontal plane in which ⃑ 𝑖 and ⃑ 𝑗 are perpendicular unit vectors. At time 𝑑 seconds ( 𝑑 β‰₯ 0 ) , the force acting on the particle is given by ⃑ 𝐹 =  ( 8 𝑑 βˆ’ 8 ) ⃑ 𝑖 + ( 4 𝑑 βˆ’ 3 ) ⃑ 𝑗  N . Find the speed of the body, 𝑣 , and its distance from the origin, 𝑑 , when 𝑑 = 3 s .

  • A 𝑣 = 7 . 5 / m s , 𝑑 = 2 . 2 5 m
  • B 𝑣 = 1 2 / m s , 𝑑 = 1 8 . 1 4 m
  • C 𝑣 = 3 2 . 8 7 / m s , 𝑑 = 2 . 2 5 m
  • D 𝑣 = 1 9 . 5 / m s , 𝑑 = 1 7 . 1 m
  • E 𝑣 = 7 . 5 / m s , 𝑑 = 1 0 1 . 6 5 m
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