Worksheet: Newton's Second Law of Motion for a Variable Mass or Force

In this worksheet, we will practice solving problems using Newton's second law for a body of variable motion and/or one that is acted upon by a variable force.

Q1:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠 = ο€Ή 6 𝑑  + 9 𝑑  m . Its mass varies with time such that π‘š = ( 8 𝑑 + 9 ) k g . Write an expression for the force acting on the body at time 𝑑 .

  • A ( 9 6 𝑑 + 1 0 8 ) N
  • B ( 3 8 4 𝑑 + 1 8 0 ) N
  • C ( 9 6 𝑑 + 1 8 0 ) N
  • D ( 1 9 2 𝑑 + 1 8 0 ) N

Q2:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠 = ο€Ή 2 𝑑 + 5 𝑑 + 4   m . Its mass varies with time such that π‘š = ( 6 𝑑 + 5 ) k g . Determine the force acting upon the body when 𝑑 = 3 s .

Q3:

A ball of mass 5 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 1 g/s. Find the magnitude of the force acting on the ball at time 𝑑 = 5 s e c o n d s , given that the displacement of the ball is expressed by the relation s c ( 𝑑 ) = ο€Ό 2 3 𝑑 + 𝑑 + 7 𝑑 + 1    , where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q4:

A body of mass 6 kg is initially at rest at a point 𝑂 . It starts to move under the action of a force 𝐹 ( π‘₯ ) = ( 2 π‘₯ + 8 ) N where π‘₯ m is the displacement of the body from 𝑂 . Find the displacement of the body when its velocity is 𝑣 = 4 / m s .

Q5:

A body of mass 2 kg moves along a straight line under the action of a force, 𝐹 . The force acting on the body is 𝐹 = ( 9 π‘₯ + 7 ) N, where π‘₯ is the displacement of the body from its initial position. Determine velocity of the body when π‘₯ = 4 m .

Q6:

A body moves in a straight line under the action of a force, 𝐹 = ο€Ό 2 5 𝑣 + 3  N , where 𝑣 is the velocity of the body at time 𝑑 seconds. If the initial velocity of the body is 3 m/s, at what time does the body reach a speed of 7 m/s?

Q7:

The mass of a body at time 𝑑 is given by π‘š ( 𝑑 ) = ( 2 𝑑 + 1 2 ) k g , whereas its position vector is r c ( 𝑑 ) = ο€Ή 2 𝑑 + 3 𝑑 + 1 5   , where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑 = 2 s e c o n d s .

Q8:

A metal ball of mass 220 g was moving in a straight line at 10 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.06 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑 .

  • A π‘š = 2 2 0 𝑑 + 0 . 0 6 g , 𝐹 = 0 . 6 d y n e s
  • B π‘š = 2 2 0 g , 𝐹 = 6 0 d y n e s
  • C π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 0 . 6 d y n e s
  • D π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 6 0 d y n e s

Q9:

A ball of mass 9 g was moving in a straight line on a dusty plane. The dust was accumulating on the ball’s surface at 2 g/s. The ball’s displacement is expressed by the relation s c ( 𝑑 ) = ο€Ή 𝑑 + 3 𝑑 + 7 𝑑 + 2    , where c is the unit vector in its direction of motion, 𝑑 is the time in seconds, and the magnitude of the displacement is measured in centimetres. Determine the force vector at time 𝑑 , given that its magnitude is measured in dynes.

  • A ο€Ή 1 8 𝑑 + 7 8 𝑑 + 7 2   c
  • B ( 1 2 𝑑 + 1 2 ) c
  • C ( 5 4 𝑑 + 5 4 ) c
  • D ο€Ή 1 8 𝑑 + 7 8 𝑑 + 6 8   c
  • E ο€Ή 6 𝑑 + 3 9 𝑑 + 6 8 𝑑 + 6 3    c

Q10:

A body of mass 20 kg started moving from rest along the π‘₯ -axis. When the body’s displacement relative to the origin was 𝑠 m in the direction of π‘₯ increasing, it moved under the effect of a force given by 𝐹 = ( 1 0 𝑠 + 5 ) N . Find the body’s velocity at 𝑠 = 8 m .

Q11:

A body of mass 11 kg started moving along the π‘₯ -axis at an initial velocity 8 m/s in the direction of π‘₯ increasing. After time 𝑑 seconds, where 𝑑 β‰₯ 0 , the body’s velocity was 𝑣 m/s in the same direction, and the force acting on the body was 𝐹 = ο€Ό 4 6 𝑣 + 6  N . Determine the value of 𝑑 at which 𝑣 = 1 0 / m s .

Q12:

A ball of mass 22 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 2 g/s. Find the magnitude of the force acting on the ball at time 𝑑 = 2 s e c o n d s , given that the displacement of the ball is expressed by the relation s c ( 𝑑 ) = ο€Ό 1 3 𝑑 + 2 𝑑 + 𝑑 + 8    , where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q13:

The mass of a body at time 𝑑 is given by π‘š ( 𝑑 ) = ( 3 𝑑 + 1 0 ) k g , whereas its position vector is r c ( 𝑑 ) = ο€Ή 2 𝑑 + 2 𝑑 + 1 3   , where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑 = 2 s e c o n d s .

Q14:

A metal ball of mass 104 g was moving in a straight line at 6 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.08 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑 .

  • A π‘š = 1 0 4 𝑑 + 0 . 0 8 g , 𝐹 = 0 . 4 8 d y n e s
  • B π‘š = 1 0 4 g , 𝐹 = 4 8 d y n e s
  • C π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 0 . 4 8 d y n e s
  • D π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 4 8 d y n e s

Q15:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠 = ο€Ή 3 𝑑 + 8 𝑑 + 7   m . Its mass varies with time such that π‘š = ( 5 𝑑 + 4 ) k g . Determine the force acting upon the body when 𝑑 = 4 s .

Q16:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠 = ο€Ή 9 𝑑  + 7 𝑑  m . Its mass varies with time such that π‘š = ( 5 𝑑 + 8 ) k g . Write an expression for the force acting on the body at time 𝑑 .

  • A ( 9 0 𝑑 + 1 4 4 ) N
  • B ( 3 6 0 𝑑 + 1 7 9 ) N
  • C ( 9 0 𝑑 + 1 7 9 ) N
  • D ( 1 8 0 𝑑 + 1 7 9 ) N

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