The projection of the Newton law for a ball moving in an italic direction

Question

if I want to push the bowling ball in a direction which is not parallel to the lane, then I need to make a Projection on the axes (the x axis is parallel with the width of the lane and the y axis is parallel with the length of the lane) ... using the Newton's laws of motion
$sum vec F = m vec a$ ,when I projected the force(which is the friction force) and the acceleration I got this:
$$ -F_x   cos(theta) = mcdot a_xcdot cos(theta)quad text{ and } -F_ycdot sin(theta) = mcdot a_ycdot  sin(theta)$$
by canceling out what is similar it gave me the same relation if the ball was moving forward with no angle
$$ -F_x  = mcdot  a_x quad text{and} quad  -F_y = mcdot  a_y$$
why did I get that ?

Steeven · Accepted Answer

You have the $cos(theta)$ and $sin(theta)$ doubled.
Start by simply ackowledging that you need the force component along the x-axis without plugging anything in yet:
$$F_x = ma_x.$$
Next, with $a_x$ being the wanted unknown and $F_x$ being a needed unknown, find an expression for $F_x$. Such an expression might be set up trigonometrically as $cos(theta)=frac{F_x}{F}$:
$$Fcos(theta) = ma_x.$$
Note how this includes $F$ and not $F_x$. And we here only have one cosine term. From this you can easily solve for $a_x$.
Same approach along the y-axis and then you find $a_y$, finalising you acceleration vector.

The projection of the Newton law for a ball moving in an italic direction

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