% Function Name: speedReport % Inputs (2): - (double) A vector containing the co

Question

% Function Name: speedReport
% Inputs (2): - (double) A vector containing the coefficients of a
%                         polynomial for a car's velocity at any time (in
%                         hours).
%              - (double) A number representing time (in seconds).
% Outputs (4): - (double) The position of the car at the input time
%              - (double) The velocity of the car at the input time
%              - (double) The acceleration of the car at the input time.
%              - (char) A sentence describing whether the car is slowing
%                       down, speeding up, neither slowing down or speeding
%                       up, or has stopped at that time.
%
% Function Description:
%   Psychic mathematicians have successfully theorized a polynomial
%   expression describing the velocity of someone's car at any time (in
%   hours). Given a vector containing the coefficients of a polynomial that
%   describes a car's velocity at any time (in hours) as well as a desired
%   time (in seconds), your function "speedReport" must output the
%   position, velocity, and acceleration of the car at that time and a
%   string describing state of the car in regards of its change in speed.
%   The fourth output should be determined by one of the following
%   scenarios:
%
%       1) If the car stops at that time, the second output should be: 'The
%          car has stopped.'
%       2) If the car is moving but neither speeding up nor slowing down,
%          then the output should be: 'The car is neither speeding up nor
%          slowing down.'
%       3) If the car is speeding up, the second output should be: 'The car
%          is speeding up.'
%       4) If the car is slowing down, the second output should be: 'The
%          car is slowing down.'
%
%   The first part of solving this problem in your function is determining
%   the vector of coefficients of the derivative of the velocity
%   polynomial. You should also determine the vector of coefficients of the
%   integral of the velocity polynomial; the constant that appears when
%   taking the integral should be zero. In context of the problem, the
%   acceleration of the car at any time (in hours) is the derivative of the
%   velocity. The position of the car at any time (in hours) is the
%   integral of the velocity.
%
% Notes:
%   - You are *not* allowed to use the built-in functions polyder() or
%     polyint().
%   - The input time can be a negative number.
%   - The position, velocity, and acceleration of the car at a desired time
%     can be negative.
%   - The velocity polynomial describes the car's velocity at any time in
%     hours, but the input time is given in seconds. Thus, before
%     determining the position, velocity, and acceleration output values,
%     the input time should be converted into hours first.
%
% Hints:
%   - You might find the polyval() function useful for this problem.
%   - When an object is speeding up, both its velocity and acceleration at
%     that time is of the same sign. In other words, its velocity and
%     acceleration are either both positive or both negative.
%   - When an object is slowing down, its velocity and acceleration at that
%     time is of opposite signs.
%   - When an object is neither speeding nor slowing down, its acceleration
%     at that time is zero.
%
% Test Cases:
%    vel_coef = [-2.5, 4.75, 14];
%
% [pos1, vel1, acc1, state1] = speedReport(vel_coef, 5000);
%   pos1   => 21.7932
%   vel1   => 15.7747
%   acc1   => -2.1944
%   state1 => 'The car is slowing down.'
%
% [pos2, vel2, acc2, state2] = speedReport(vel_coef, 0);
%   pos2   => 0
%   vel2   => 14
%   acc2   => 4.7500
%   state2 => 'The car is speeding up.'
%  
% [pos3, vel3, acc3, state3] = speedReport(vel_coef, -5760);
%   pos3   => -12.9067
%   vel3   => 0
%   acc3   => 12.7500
%   state3 => 'The car has stopped.'
%
% [pos4, vel4, acc4, state4] = speedReport(vel_coef, 3420);
%   pos4   => 14.7290
%   vel4   => 16.2563
%   acc4   => 0
%   state4 => 'The car is neither speeding up nor slowing down.'

Explanation / Answer

function [pos, vel, acc, state]= speedReport(vcoff,t)

t= t/(60*60);

ord=length(vcoff);

acoff=[];

pcoff=[];

for k=1:ord

p=vcoff(k)/(ord-k+1);

pcoff=[pcoff p];

a=vcoff(k)*(ord-k);

if k~=ord

acoff=[acoff a];

end

end

pcoff=[pcoff 0];

pos=polyval(pcoff,t);

vel=polyval(vcoff,t);

acc=polyval(acoff,t);

if vel==0

state='The car has stopped.';

elseif acc==0

state='The car is neither speeding up nor slowing down.';

elseif acc>0

state= 'The car is speeding up.';

elseif acc<0

state='The car is slowing down.';

end

end

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