This Homework question comes from 3 unit yr 12 Cambridge Mathematics A particle

Question

This Homework question comes from 3 unit yr 12 Cambridge Mathematics
A particle moving in simple harmonic motion has speed 12 m/s at the origin. Find the displacement-time equation if it is known that for positive constants a and n:

a) x = a cos 8t
[hint: Start by differentiating the given equation to find the equation of v. then use the fat that the speed at the origin is the maximum value of |v|

My working :
Given x = a cost 8t
v= -8a sin 8t
12 = -8a sin 8(0)
12= -8a x 0
12=0 this is where I'm stuck - the text book answer is x= 3/2 cos 8t

Explanation / Answer

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This Homework question comes from 3 unit yr 12 Cambridge Mathematics
A particle moving in simple harmonic motion has speed 12 m/s at the origin. Find the displacement-time equation if it is known that for positive constants a and n:

a) x = a cos 8t...OK
[hint: Start by differentiating the given equation to find the equation of v. then use the fat that the speed at the origin is the maximum value of |v|

My working :
Given x = a cost 8t................1...............OK
v= -8a sin 8t.....................2..............GOOD
12 = -8a sin 8(0)..............NO ......

WE ARE GIVEN DISPACEMENT EQN.AND AS YOU SAY ORIGIN IS THE MAXIMUM VALUE OF |V|

VIDE..[the origin is the maximum value of |v|]...

SO WE SHOULD FIND WHAT IS THE MAXIMUM POSIBLE |V| FROM EQN.2

AS WE KNOW SINE FUNCTION HAS A MAXIMUM VALUE OF 1 FOR ANY VALUE OF THE ANGLE .

SO MAXIMUM VALE=|V|=12=-8*A*1

A=-12/8 = -1.5

HENCE EQN. 1 BECOMES

X=-1.5COS(8T) IS THE ANSWER

WHAT IS N? YOU ARE ASKING?....Find the displacement-time equation if it is known that for positive constants a and n:

12= -8a x 0
12=0 this is where I'm stuck - the text book answer is x= 3/2 cos 8t

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