A rocket is modelled by a particle that moves along a vertical line. From launch, the rocket rises until its motor cuts out after 13 seconds. At this time it has reached a height of 490 metres above the launch pad and attained an upward velocity of 70ms−1. From this time on, the rocket has a constant upward acceleration of −10 m s−2 (due to the effect of gravity alone).
(a) Choose the s-axis (for the position of the particle that represents the rocket) to point upwards, with origin at the launch pad. Take t = 0 to be the time when the rocket motor cuts out.
(i) What is the maximum height (above the launch pad) reached by the rocket?
(ii) How long (from launch) does the rocket take to reach this maximum height?
(b) After how long (from launch) does the rocket crash on to the launch pad? Give your answer in seconds, correct to one decimal place.© BrainMass Inc. brainmass.com December 24, 2021, 11:11 pm ad1c9bdddf
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First we make a quick sketch of the rockets motion as shown below
(i) From the point of origin ( the point where the rocket motor stops firing) the rocket then is only subject to de-acceleration due to gravity so we can apply the equation  below from linear kinematics to determine the vertical distance the rocket travels before it stops its ascent. This is the maximum distance above point
Where is the velocity of the rocket at the top of its ascent (ie, it stops travelling in the vertical direction so has no vertical velocity component at the top of its ascent). Therefore, equation  becomes 
As the de-acceleration is and the initial velocity of the rocket (at the origin) is we find
The maximum height the rocket reaches is made up of the sum of the height it reaches before the motor cuts out and
Thus maximum height of the rocket
(ii) We use an equation  from linear kinematics to determine the time from the origin when the rocket reaches its maximum displacement
At the top of the ascent and at the origin so we can determine the time from 
However, this is not the time taken from launch till the rocket reaches its maximum height as we need to consider the taken by the rocket to reach the point where the rocket motor cuts out (our point of origin)
Thus the time taken for the rocket to reach its maximum height is
To determine the total time elapsed (from launch) before the rocket hits the ground we consider this as the sum of 2 parts, the time taken to reach maximum height which was calculated in part (a(ii)) and the time taken to drop just under the influence of gravity
We know the maximum height as determined in part (a(i)) so denote this as
We also know that at maximum height just before the rocket starts to drop the initial vertical component of velocity is
We therefore use equation  from linear kinematics to determine the time for the rocket to drop to ground
As , (an acceleration now as it drops to ground) and equation  becomes
Hence total time for the rocket to reach the ground from launch is© BrainMass Inc. brainmass.com December 24, 2021, 11:11 pm ad1c9bdddf>