SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Mechanics, EXT2 M1 2022 HSC 15a

A machine is lifted from the floor of a room using two ropes. The two ropes ensure that the horizontal components of the forces are balanced at all times. It is assumed that at all times the machine moves vertically upwards at a constant velocity.

The machine is located in a room with height metres.

One of the ropes is attached to the point on the machine and to the fixed point on the ceiling of the room. The point is a distance metres to the left of . Let the vertical distance from to the ceiling be metres and let be the angle this rope makes with the horizontal.

The other rope is attached to the point and to the fixed point on the floor of the room. The point is a distance metres to the right of . Let be the angle this rope makes with the horizontal.

Let the tension in the first rope be newtons, the tension in the second rope be newtons, the mass of the machine be kilograms and the acceleration due to gravity be .
 

  1. By considering horizontal and vertical components of the forces at , show that
  2.               (3 marks)

  3. Hence, or otherwise, show that the point cannot be lifted to a position metres above the floor.  (2 marks)

Show Answers Only
Show Worked Solution

i.   
       


 

 
   

 


Mean mark (i) 56%.

ii.   

 
   

 

 
 
 
 
 
 

 


♦♦♦ Mean mark (ii) 27%.

Mechanics, EXT2 M1 2022 HSC 12c

A particle with mass 1 kg is moving along the -axis. Initially, the particle is at the origin and has speed m s-1 to the right. The particle experiences a resistive force of magnitude    newtons, where m s-1 is the speed of the particle after seconds. The particle is never at rest.

  1. Show that    (1 mark)

  2. Hence, or otherwise, find as a function of .  (2 marks)

Show Answers Only
Show Worked Solution

i.   

 
 
 

  

ii.  
 
   

 

 
   

Mechanics, EXT2 M1 2020 HSC 12a

A 50-kilogram box is initially at rest. The box is pulled along the ground with a force of 200 newtons at an angle of 30° to the horizontal. The box experiences a resistive force of    newtons, where is the normal force, as shown in the diagram.
 
Take the acceleration due to gravity to be 10m/s2.
 

  1. By resolving the forces vertically, show that  .   (2 marks)

  2. Show that the net force horizontally is approximately 53.2 newtons.  (2 marks)

  3. Find the velocity of the box after the first three seconds.  (2 marks)

Show Answers Only
Show Worked Solution

i.   

 

ii.   

 
 
 

 

iii.   
 
 

 

 
 

Mechanics, EXT2 M1 EQ-Bank 2

A particle with mass moves horizontally against a resistance force , equal to    where is the particle's velocity.

Initially, the particle is travelling in a positive direction from the origin at velocity .

  1. Show that the particle's displacement from the origin, , can be expressed as
     
           (2 marks)

     

  2. Show that the time, , when the particle is travelling at velocity , is given by
     
           (4 marks)

     

  3. Express as a function of , and hence find the limiting values of and .  (2 marks)

Show Answers Only
  2. `text(See Worked Solutions)“
Show Worked Solution

i.   

 

 


 

 
 

 

ii.   


 

     
     

 

 
 
 
 

 


 

 
 

 

iii.
 
 
 
 
 

 

Mechanics, EXT2 M1 2011 HSC 6a

Jac jumps out of an aeroplane and falls vertically. His velocity at time    after his parachute is opened is given by  , where    and    is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is  , where    is a positive constant. Let    be Jac’s mass and    the acceleration due to gravity. Jac’s terminal velocity with the parachute open is  

Jac’s equation of motion with the parachute open is

   (Do NOT prove this.)

  1. Explain why Jac’s terminal velocity    is given by  
     
           (1 mark)

  2. By integrating the equation of motion, show that    and    are related by the equation
     
           (3 marks)

  3. Jac’s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes.

     

    Jac opens his parachute when his speed is   Gil opens her parachute when her speed is   Jac’s speed increases and Gil’s speed decreases, both towards  

     

    Show that in the time taken for Jac's speed to double, Gil's speed has halved.  (3 marks)

Show Answers Only
Show Worked Solution

i.  

  

ii. 

  ­

  

♦ Mean mark part (ii) 39%.

­
 
 
 
 
 

 

 

iii. 

 
 
 

 

 

♦♦♦ Mean mark part (iii) 16%.

 
 
 

 

Mechanics, EXT2 M1 2013 HSC 15d

A ball of mass is projected vertically into the air from the ground with initial velocity  . After reaching the maximum height it falls back to the ground. While in the air, the ball experiences a resistive force , where is the velocity of the ball and is a constant.

The equation of motion when the ball falls can be written as

      (Do NOT prove this.)

  1. Show that the terminal velocity   of the ball when it falls is
  2.      (1 mark)

  3. Show that when the ball goes up, the maximum height    is
  4.      (3 marks)

  5. When the ball falls from height    it hits the ground with velocity  .
  6. Show that    (2 marks)

Show Answers Only
Show Worked Solution

i. 

 

ii. 

♦ Mean mark 43%.

MARKER’S COMMENT: More than half of students incorrectly wrote the equation to solve as
 
 
 

 

 

iii. 

♦♦ Mean mark 14%.
 

 

 
 
 
 

 

Mechanics, EXT2 M1 2014 HSC 14c

A high speed train of mass starts from rest and moves along a straight track. At time hours, the distance travelled by the train from its starting point is km, and its velocity is km/h.

The train is driven by a constant force    in the forward direction. The resistive force in the opposite direction is  , where    is a positive constant. The terminal velocity of the train is 300 km/h.

  1. Show that the equation of motion for the train is
     
         .  (2 marks)

  2. Find, in terms of    and  , the time it takes the train to reach a velocity of 200 km/h.  (4 marks)

Show Answers Only
Show Worked Solution

i.   
 

 

ii.