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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 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 seconds, given that the displacement of the ball is expressed by the relation s ( 𝑑 ) = ο€Ό 2 3 𝑑 + 𝑑 + 7 𝑑 + 1  𝑐 3 2 , where 𝑐 is a unit vector in the direction of the motion and the displacement is measured in centimeters.

  • A288 dynes
  • B22 dynes
  • C110 dynes
  • D287 dynes
  • E670 dynes

Q2:

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 seconds, given that the displacement of the ball is expressed by the relation s ( 𝑑 ) = ο€Ό 1 3 𝑑 + 2 𝑑 + 𝑑 + 8  𝑐 3 2 , where 𝑐 is a unit vector in the direction of the motion and the displacement is measured in centimeters.

  • A250 dynes
  • B16 dynes
  • C176 dynes
  • D234 dynes
  • E338 dynes

Q3:

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 .

Q4:

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 , where is the unit vector in its direction of motion, is the time in seconds, and the magnitude of the displacement is measured in centimetres. Determine the force vector at time , given that its magnitude is measured in dynes.

  • A
  • B
  • C
  • D
  • E

Q5:

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  2 , where c is a constant one 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 seconds.

  • A 68 N
  • B 64 N
  • C 50 N
  • D 86 N

Q6:

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  2 , where c is a constant one 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 seconds.

  • A 80 N
  • B 64 N
  • C 74 N
  • D 94 N

Q7:

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 .

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 dynes
  • B π‘š = 2 2 0 g , 𝐹 = 6 0 dynes
  • C π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 0 . 6 dynes
  • D π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 6 0 dynes

Q9:

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 dynes
  • B π‘š = 1 0 4 g , 𝐹 = 4 8 dynes
  • C π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 0 . 4 8 dynes
  • D π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 4 8 dynes

Q10:

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 .

Q11:

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 .

Q12:

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 .

Q13:

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?

Q14:

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

Q15:

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 + 2 𝑑   m . Its mass varies with time such that π‘š = ( 5 𝑑 + 3 ) k g . Write an expression for the force acting on the body at time 𝑑 .

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

Q16:

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 .

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