5.5 Kinematics

A particle moves along a horizontal line starting at the point . The displacement-time graph for the first  seconds of its motion is shown below. Displacement is measured in metres.

(i)      Write down the displacement of the particle after 2 seconds.

(ii)     Write down the displacement of the particle after 4 seconds.

Find the velocity of the particle between 13 and 20 seconds.

Find the speed of the particle between 7 and 10 seconds.

Find the total distance travelled by the particle after 20 seconds.

A cricket ball is projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

where  metres is the height of the cricket ball above ground level after  seconds.

Find the times at which the cricket ball is exactly 3 m above the ground.

For how long is the cricket ball at least 3 m above the ground?

A player catches the cricket ball (on its way down) at a height of 0.8 m above the ground.

Find the length of time the ball was in the air.

Find the total distance travelled by the ball.

Find the velocity of the cricket ball at  second.

A softball is thrown upwards from the top of a 10 m tall building.
The height,  m of the ball above the ground after  seconds is modelled by the function

Write down the value of .

Find the height of the ball after 2 seconds.

Find the time at which the ball is at the same height as it was when thrown.

Find the time the ball first hits the ground.

Find  and hence show that the acceleration at any time is .

A particle moves along a straight line with a velocity,   given by     where  is measured in seconds such that  

Find the acceleration of the particle at time .

State the time when the particle comes to rest.

Find the total distance travelled by the particle.

A particle is found to have an acceleration,, according to the function

               ,where 

Find an expression for the velocity, , of the particle given that  . 

Find the velocity of the particle at .

A particle, moving in a straight line, is found to have a velocity  where  is measured in and  time  is measured in seconds such that 

Find the time(s) when the particle is instantaneously at rest.

Find the time(s) when the particle changes direction.

Find the distance travelled in the first second of motion.

Find the acceleration of the particle at the instant it first changes direction.

Find the displacement of the particle from its starting point to the point when 

A particle is moving along a straight line.  The position of the particle at time  seconds, measured in metres relative to a fixed origin point, is denoted by  .

The particle starts at the origin at time  ,  and its motion over the next eight seconds is described by the equation

Find an expression for .

Hence determine the maximum distance of the particle from the origin during the first eight seconds of its movement.

Find the change in displacement of the particle during the first eight seconds of its movement.

Find the total distance travelled by the particle during the first eight seconds of its movement.

Find an expression for the particle’s acceleration .

A particle is moving along a straight line.  The position of the particle at any given time, measured in metres relative to a fixed origin point, is denoted by .

It is known that the velocity, , of the particle is dependent on the particle’s position, and that the velocity may be described by the equation

Use the chain rule to explain why the acceleration, , of the particle may be expressed in the form 

Show that the derivative of    is  .

Hence find an expression for the acceleration of the particle in terms of , being sure to indicate the domain of  values for which the expression is valid.

Identify the minimum and maximum values of

          (i)     the speed of the particle

          (ii)    the magnitude of the particle’s acceleration

along with the values of  for which those occur.

The level of water, m, in an estuary relative to the mean sea level, is observed over a  24-hour period starting from midnight and is modelled by the function

where  is the time in hours after midnight.

Find the rate of change of the height of the water level at 7 am.

Find the percentage of time within the 24-hour period that the water level remains above the mean sea level.

The scientists observing the water level in the estuary anchor a buoy in place such that its horizontal position is fixed, but it is able to move up and down vertically with the changing water level

Find the total vertical distance that the buoy moves during the 24-hour time period.

For a particle P moving in a straight line let    for   ,  where  is the velocity of the particle and  is the elapsed time in seconds.

Sketch the graph of  on the grid below.

Find the times at which the points of inflection would occur on a displacement-time graph representing the particle’s movement, and explain the significance of these points in the context of this question.

Hence find the values of  for which the displacement-time graph would be concave down.

Two particles,  and ,  are observed moving along a straight line.  The displacements of the particles, respectively  and , in metres relative to a fixed point O can be modelled for  by the following functions

where  is the time in seconds from the start of the observation.

Find an expression for the distance between the two particles at time .

Hence find

(i)       the maximum distance of the particles from one another

A collision occurs between the particles during the time of observation.

Find the velocity of each of the particles 0.5 seconds before the time that they collide.

A particle starts from point X and moves in a straight line. The graph below shows its velocity, ms^-1 after  seconds for .

The particle has an instantaneous velocity of  ms^-1 at  and   .

The function  represents the displacement of the particle from point X after seconds.

It is known that the particle travels 22 metres in the first 3 seconds.

It is also known that  and .

Find the value of .

Find the total distance travelled by the particle in the first 7 seconds.

A particle P moves along a straight line.  The velocity of the particle after  seconds,  is given by

Write down the first value of at which P changes its direction of motion.

Find the total distance travelled by P during the periods when its speed is increasing.

A second particle, Q, also moves along a straight line.  Its velocity after  seconds, is given by 

After  seconds, the total distance that Q has travelled is the same as the distance that P travels during its periods of increasing speed.

Find the value of .

A particle moves in a straight line starting from point P.  The particle is found to have a velocity,   ms^-1, given by the piecewise function

Find the maximum velocity reached by the particle and the time at which that maximum velocity is reached.

Find an expression for the displacement of the particle from the starting point P at time , given that the displacement of the particle from point P at the end of the time period is  m.

Find the total distance travelled by the particle.

A particle A moves along a horizontal straight line .  The displacement,  cm, of particle A from a fixed point P on  is given by the function

where  is the time in seconds from the start of the motion.

Starting at the same time, another particle, B, moves along a horizontal straight line  which is parallel to .

The velocity of particle B, , at time  seconds is given by

Find the value(s) of for which particle A is at point P.

Find the value of at which particle A first changes direction.

Find the total distance travelled by particle A in the first 8 seconds of its motion.

The displacement,  cm, of particle B is measured relative to a fixed point Q on .

Given that , find:

the displacement function   for particle B

A particle is moving along a straight line.  The position of the particle at time  seconds, measured in metres relative to a fixed origin point, is denoted by  .

The particle starts at the origin at time    with a velocity of  ,  and its motion over the next ten seconds is described by the equation

Considering the total distance travelled by the particle over the ten seconds, calculate the percentage of that total distance that the particle travels in the first five seconds of its movement.

Find the greatest distance from the origin point that the particle reaches, and the time  at which that greatest distance is reached.

Professor Goodwin Vundera, a kinematics researcher, has been studying a particular type of particle.  The particle is known only to move along a straight line, with its acceleration, , and velocity, , both dependent on the particle’s displacement,  m, with reference to a fixed origin point.

The professor has defined a new real-valued function, the ‘Vundera function’, which he believes captures all necessary information about the motion of the particle.  This Vundera function, , is defined by   ,  where   and  are functions describing, respectively, the particle’s acceleration and velocity in terms of .  For one of the particles studied by the professor, the associated Vundera function is found to be

where  and  are real constants.

Given that  has the largest possible domain, write down the values of  and .

Additionally it is known that when , the velocity of the particle is .

By using the above information and solving an appropriate indefinite integral, find expressions for the functions  and  associated with the particle.

Explain the relationship between the particle’s speed and acceleration as  varies across all the values in the domain.