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

In this worksheet, we will practice applying Newton’s second law when the forces acting on a body and the motion caused by them are represented in the vector notation.

Q1:

If a body of mass 1 kg moves under the action of forces F i j k = ( + 8 5 ) N and F i j k = ( 2 7 + 8 ) N , what is its acceleration?

  • A ( 3 + 2 + 3 ) i j k m/s2
  • B ( 6 + + 3 ) i j k m/s2
  • C ( 3 + + 9 ) i j k m/s2
  • D ( 3 + + 3 ) i j k 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 and 𝑣 = 9 . 8 𝑡 + 1 2 where 𝑣 and 𝑣 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 .

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

Q3:

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

  • A 𝑎 = 6 , 𝑏 = 8
  • B 𝑎 = 2 , 𝑏 = 8
  • C 𝑎 = 2 , 𝑏 = 0
  • D 𝑎 = 1 0 , 𝑏 = 1 6

Q4:

A particle of mass 𝑚 kg is moving under the action of two forces: F i j = 8 𝑚 + 6 𝑚 and F i = 4 𝑚 , where i and j are two perpendicular unit vectors. Find the acceleration a of the particle and its magnitude | | a in metres per second squared.

  • A a i j = 1 2 6 , | | = 6 5 / a m s
  • B a i j = 4 + 6 , | | = 2 1 3 / a m s
  • C a i j = 1 2 + 6 , | | = 6 3 / a m s
  • D a i j = 1 2 + 6 , | | = 6 5 / a m s

Q5:

If the forces F i j k = ( 𝑥 + 𝑦 + 𝑧 ) N and F i j k = ( 5 6 3 ) N acting on a body of mass 6 kg, cause an acceleration a i j k = ( 5 + 2 4 ) / m s , what are the values of 𝑥 , 𝑦 , and 𝑧 ?

  • A 𝑥 = 3 0 , 𝑦 = 1 2 , 𝑧 = 2 4
  • B 𝑥 = 2 5 , 𝑦 = 6 , 𝑧 = 2 7
  • C 𝑥 = 0 , 𝑦 = 4 , 𝑧 = 7
  • D 𝑥 = 3 5 , 𝑦 = 1 8 , 𝑧 = 2 1

Q6:

Given that the motion of a body of mass 2 kg is represented by the relation r c ( 𝑡 ) = 6 𝑡 + 1 5 𝑡 + 2 , where c is a constant unit vector, r is measured in metres, and 𝑡 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 i j = 𝑎 + 𝑏 , where i and j are two orthogonal unit vectors. If the displacement vector of the body at time 𝑡 is given by s i j ( 𝑡 ) = ( 9 𝑡 ) + ( 𝑡 + 3 ) , find 𝑎 and 𝑏 .

  • A 𝑎 = 9 , 𝑏 = 2
  • B 𝑎 = 2 , 𝑏 = 1 8
  • C 𝑎 = 1 8 , 𝑏 = 1
  • D 𝑎 = 1 8 , 𝑏 = 2
  • E 𝑎 = 9 , 𝑏 = 1

Q8:

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

  • A 𝑎 = 1 0 , 𝑏 = 1
  • B 𝑎 = 1 0 , 𝑏 = 1
  • C 𝑎 = 1 0 , 𝑏 = 1
  • D 𝑎 = 1 0 , 𝑏 = 1

Q9:

A body of mass 9 g was moving on a plane under the effect of the force F i j = ( 1 0 ) dynes. Given that the position vector of the body is given by the relation r i j ( 𝑡 ) = 𝑎 𝑡 + 7 + 𝑏 𝑡 + 6 𝑡 c m , determine 𝑎 and 𝑏 .

  • A 𝑎 = 1 2 , 𝑏 = 5
  • B 𝑎 = 1 1 8 , 𝑏 = 3 2 9
  • C 𝑎 = 3 2 9 , 𝑏 = 5 9
  • D 𝑎 = 1 1 8 , 𝑏 = 5 9

Q10:

A body of mass 7 kg moves under the action of three forces, F i j = ( 𝑎 + 3 ) N , F i j = ( 6 6 ) N , and F i j = ( 6 + 𝑏 ) N . Given that the displacement of the body at time 𝑡 seconds is s i j = 𝑡 + 6 + 5 𝑡 + 5 m , determine the values of 𝑎 and 𝑏 .

  • A 𝑎 = 1 4 , 𝑏 = 7 9
  • B 𝑎 = 1 0 , 𝑏 = 1 3
  • C 𝑎 = 1 4 , 𝑏 = 6 1
  • D 𝑎 = 2 , 𝑏 = 7 3

Q11:

A particle of unit mass is moving such that its velocity at a given time 𝑡 is represented by v i ( 𝑡 ) = 8 𝑎 𝑡 + 5 𝑏 𝑡 , where i is a constant unit vector. Given that the force acting on the particle at time 𝑡 is F i ( 𝑡 ) = ( 1 0 𝑡 + 4 ) , find 𝑎 and 𝑏 .

  • A 𝑎 = 5 8 , 𝑏 = 4 5
  • B 𝑎 = 5 8 , 𝑏 = 4 5
  • C 𝑎 = 5 8 , 𝑏 = 4 5
  • D 𝑎 = 5 8 , 𝑏 = 4 5

Q12:

A particle of unit mass is moving along a certain path, where its velocity at time 𝑡 is given by the relation v i = 𝑎 𝑡 + 𝑏 𝑡 , where i is a constant unit vector. Given that the force acting on the particle is constant and given by the relation F i = 9 1 , determine the values of the constants 𝑎 and 𝑏 .

  • A 𝑎 = 0 , 𝑏 = 9 1
  • B 𝑎 = 9 1 , 𝑏 = 0
  • C 𝑎 = 9 1 , 𝑏 = 0
  • D 𝑎 = 0 , 𝑏 = 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 i j = ( 9 𝑡 + 3 ) + 9 𝑡 , where i and j are perpendicular unit vectors, find the displacement in terms of 𝑡 .

  • A 6 𝑡 + 1 2 𝑡 + 6 𝑡 i j
  • B 1 2 𝑡 + 6 𝑡 + 6 𝑡 i j
  • C 6 𝑡 + 6 𝑡 + 1 8 𝑡 i j
  • D 6 𝑡 + 6 𝑡 + 6 𝑡 i j

Q14:

A particle of mass 5 kg was in motion. The components of its velocity in the horizontal and vertical directions were 𝑣 = 3 / m s and 𝑣 = ( 4 . 7 𝑡 + 1 4 ) / m s , respectively. Determine the magnitude, 𝑣 , and direction, 𝜃 , of its initial velocity and the force F acting on it.

  • A 𝑣 = 2 3 / m s , 𝜃 = 7 2 7 , F j = 4 . 7
  • B 𝑣 = 1 9 9 / m s , 𝜃 = 7 2 7 , F j = 2 3 . 5
  • C 𝑣 = 2 0 5 / m s , 𝜃 = 7 7 5 4 , F j = 4 . 7
  • D 𝑣 = 2 0 5 / m s , 𝜃 = 7 7 5 4 , F j = 2 3 . 5

Q15:

A body of mass 𝑚 is moving under the action of a force F . Its velocity at time 𝑡 seconds is given by the relation v i ( 𝑡 ) = ( 6 𝑎 𝑡 + 𝑏 ) / m s , where i is a unit vector in the direction of its motion, and 𝑎 and 𝑏 are constants. Given that the initial velocity of the body v i = 1 5 / m s and F i = ( 1 2 𝑚 ) N , find the body’s speed at 𝑡 = 1 4 s e c o n d s .

Q16:

Three forces, F i j k = ( 𝑎 + 4 9 ) N , F i j k = ( 3 8 + 𝑐 ) N , and F i j k = ( 4 + 𝑏 + 8 ) N , where i , j , and k are three perpendicular unit vectors, are acting upon a body of unit mass. If the displacement vector of the particle is s i j k = ( 4 𝑡 ) + 6 𝑡 + 3 𝑡 + 8 𝑡 + 7 m , determine the constants 𝑎 , 𝑏 , and 𝑐 .

  • A 𝑎 = 7 , 𝑏 = 1 8 , and 𝑐 = 1 7
  • B 𝑎 = 7 , 𝑏 = 1 6 , and 𝑐 = 1
  • C 𝑎 = 1 , 𝑏 = 1 6 , and 𝑐 = 1 7
  • D 𝑎 = 7 , 𝑏 = 1 6 , and 𝑐 = 1 7
  • E 𝑎 = 1 , 𝑏 = 1 0 , and 𝑐 = 9

Q17:

A body of mass 478 g has an acceleration of ( 4 + 3 ) i 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 i j = ( 8 8 ) / m s , where i and j are two perpendicular unit vectors. The force F i j = ( 4 5 ) N acted on the body for 8 seconds. Find the body’s speed after the action of this force.

  • A 8 4 1 m/s
  • B 8 6 1 m/s
  • C 40 m/s
  • D 2 4 5 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 𝑡 seconds ( 𝑡 0 ) , the force acting on the particle is given by F i j = [ ( 8 𝑡 8 ) + ( 4 𝑡 3 ) ] N . Find the speed of the body, 𝑣 , and its distance from the origin, 𝑑 , when 𝑡 = 3 s .

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

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