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Mechanics, EXT2 M1 2024 HSC 13c

A particle of unit mass moves horizontally in a straight line. It experiences a resistive force proportional to , where m s is the speed of the particle, so that the acceleration is given by  .

Initially the particle is at the origin and has a velocity of 40 m s to the right. After the particle has moved 15 m to the right, its velocity is 10 m s (to the right).

  1. Show that  .   (3 marks)

  2. Show that  .   (1 mark)

  3. At what time will the particle's velocity be 30 m s to the right?   (3 marks)

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Mechanics, EXT2 M1 2023 HSC 14c

A projectile of mass kg is launched vertically upwards from the origin with an initial speed m s. The acceleration due to gravity is ms.

The projectile experiences a resistive force of magnitude newtons, where is a positive constant and is the speed of the projectile at time seconds.

  1. The maximum height reached by the particle is metres.
  2. Show that  .  (3 marks)

  3. When the projectile lands on the ground, its speed is , where is less than the magnitude of the terminal velocity.
  4. Show that  .  (3 marks)

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♦♦ Mean mark (ii) 31%.

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)

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Mechanics, EXT2 M1 2022 HSC 8 MC

As a projectile of mass kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to the square of the speed of the projectile. The constant of proportionality is the positive number . The position of the particle at time is denoted by . The acceleration due to gravity is .

Based on Newton's laws of motion, which equation models the motion of this projectile?

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♦♦♦ Mean mark 26%.

Mechanics, EXT2 M1 2021 HSC 14b

An object of mass 5 kg is on a slope that is inclined at an angle of 60° to the horizontal. The acceleration due to gravity is   and the velocity of the object down the slope is .

As well as the force due to gravity, the object is acted on by two forces, one of magnitude newtons and one of magnitude newtons, both acting up the slope.

  1. Show that the resultant force down the slope is    newtons.  (2 marks)

  2. There is one value of such that the object will slide down the slope at a constant speed.
  3. Find this value of in , correct to 1 decimal place, given that  .  (2 marks)

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Mechanics, EXT2 M1 2021 HSC 15c

An object of mass 1 kg is projected vertically upwards with an initial velocity of m/s. It experiences air resistance of magnitude newtons where is the velocity of the object, in m/s, and is a positive constant. The height of the object above its starting point is metres. The time since projection is seconds and acceleration due to gravity is m/s².

  1. Show that the time for the object to reach its maximum height is    seconds.  (3 marks)

  2. Find an expression for the maximum height reached by the object, in terms of , , and .  (3 marks)

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Mean mark part (ii) 54%.
 

 

 
 

Mechanics, EXT2 M1 EQ-Bank 4

A torpedo with a mass of 80 kilograms has a propeller system that delivers a force of on the torpedo, at maximum power. The water exerts a resistance on the torpedo proportional to the square of the torpedo's velocity .

  1. Explain why 
     
    where is a positive constant.  (1 mark)

  2. If the torpedo increases its velocity from    to  , show that the distance it travels in this time, , is given by
     
           (3 marks)

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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)

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Mechanics, EXT2 M1 EQ-Bank 1

A canon ball of mass 9 kilograms is dropped from the top of a castle at a height of metres above the ground.

The canon ball experiences a resistance force due to air resistance equivalent to  , where is the speed of the canon ball in metres per second. Let    and the displacement, metres at time seconds, be measured in a downward direction.

  1. Show the equation of motion is given by
     
           (1 mark)

  2. Show, by integrating using partial fractions, that
     
           (5 marks)

     

  3. If the canon hits the ground after 4 seconds, calculate the height of the castle, to the nearest metre.  (3 marks)

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Mechanics, EXT2 M1 2018 HSC 14b

A falling particle experiences forces due to gravity and air resistance. The acceleration of the particle is  , where and are positive constants and is the speed of the particle. (Do NOT prove this.)

Prove that, after falling from rest through a distance, , the speed of the particle will be  .  (3 marks)

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Mechanics, EXT2 M1 2017 HSC 13c

A particle is projected upwards from ground level with initial velocity  , where is the acceleration due to gravity and is a positive constant. The particle moves through the air with speed    and experiences a resistive force.

The acceleration of the particle is given by  . Do NOT prove this.

The particle reaches a maximum height, , before returning to the ground.

Using  , or otherwise, show that    metres.  (4 marks)

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Mechanics, EXT2 M1 2015 HSC 15a

A particle   of unit mass travels horizontally through a viscous medium. When  , the particle is at point    with initial speed  . The resistance on particle    due to the medium is  , where    is the velocity of the particle at time    and    is a positive constant.

When  , a second particle    of equal mass is projected vertically upwards from    with the same initial speed    through the same medium. It experiences both a gravitational force and a resistance due to the medium. The resistance on particle    is  , where    is the velocity of the particle    at time  . The acceleration due to gravity is  .

  1. Show that the velocity    of particle    is given by  
     
           (2 marks)

  2. By considering the velocity    of particle  , show that
     
           (3 marks)

  3. Show that the velocity    of particle    when particle    is at rest is given by
     
           (1 mark)

  4. Hence, if    is very large, explain why  
     
           (1 mark)

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Mechanics, EXT2 M1 2007 HSC 6b

A raindrop falls vertically from a high cloud. The distance it has fallen is given by

where is in metres and is the time elapsed in seconds.

  1. Show that the velocity of the raindrop, metres per second, is given by
     
           (2 marks)

  2. Hence show that    (2 marks)

  3. Hence, or otherwise, show that    (2 marks)

  4. The physical significance of the 9.8 in part (iii) is that it represents the acceleration due to gravity.

     

    What is the physical significance of the term    (1 mark)

  5. Estimate the velocity at which the raindrop hits the ground.   (1 mark)

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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)

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♦ Mean mark part (ii) 39%.

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♦♦♦ Mean mark part (iii) 16%.

 
 
 

 

Mechanics, EXT2 M1 2012 HSC 13a

An object on the surface of a liquid is released at time    and immediately sinks. Let    be its displacement in metres in a downward direction from the surface at time    seconds.

The equation of motion is given by

,

where    is the velocity of the object.  

  1. Show that  .   (4 marks)

  2. Use   to show that  
     
            (2 marks)

     

  3. How far does the object sink in the first 4 seconds?   (2 marks)

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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)

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♦ Mean mark 43%.

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

 

 

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♦♦ 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)

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