Assistance needed with vector velocity problem, please

[SelHype]

An asteroid is discovered heading straight toward Earth at 15 km/s. An international team manages to attach a giant rocket engine to the asteroid. The rocket fires for 10 min, after which the asteroid is moving at 28$$\circ$$ to its original path at a speed of 19 km/s.

Find its average acceleration (a_x, a_y) in m/s².

I first began by using the equation a² = b² + c² -2bc(cos$$\alpha$$) where b is 15 km/s and c is 19 km/s.

a² = 225 + 361 - 570(cos28$$\circ$$)
a² = 82.7 km/s
9.1 km/s m= $$\Delta$$v

a= 9.1 / 600 = .0152 km/s² = 15.2 m/s² The answer is $$r\hat{}$$ = (3.0$$i\hat{}$$ + 15 $$j\hat{}$$) m/s².

I am unsure as to whether or not I have done this correctly because I do not know where to go from here. My professor gave use this hint for this problem:

The asteroid is initially going in the +x direction! From the given initial and final
velocities, find $$\Delta$$v_x and $$\Delta$$ v_y. Use a_x = $$\Delta$$v_x/$$\Delta$$t and a_y = $$\Delta$$v_y/$$\Delta$$t

 

[tiny-tim]

Welcome to PF!

An asteroid is discovered heading straight toward Earth at 15 km/s. An international team manages to attach a giant rocket engine to the asteroid. The rocket fires for 10 min, after which the asteroid is moving at 28$$\circ$$ to its original path at a speed of 19 km/s.

Find its average acceleration (a_x, a_y) in m/s².

I first began by using the equation a² = b² + c² -2bc(cos$$\alpha$$) where b is 15 km/s and c is 19 km/s.

a² = 225 + 361 - 570(cos28$$\circ$$)
a² = 82.7 km/s
9.1 km/s m= $$\Delta$$v

a= 9.1 / 600 = .0152 km/s² = 15.2 m/s²

The answer is $$r\hat{}$$ = (3.0$$i\hat{}$$ + 15 $$j\hat{}$$) m/s².

Hi SelHype! Welcome to PF!

The question asks for (a_x, a_y).

Your cosine formula only gave you the magnitude, |a| (which was correct ) …

but you won't get the direction without using the sine formula also, which is far too long-winded a method.

There are two ways of dealing with vectors … the good old trigonometry way that the ancient Greeks would have used, and the coordinate method.​

You've used the slow ancient Greek way.

Your professor wants you to use the quicker coordinate way.

Do what your professor suggested …

 

[SelHype]

Hi SelHype! Welcome to PF!

The question asks for (a_x, a_y).

Your cosine formula only gave you the magnitude, |a| (which was correct ) …

but you won't get the direction without using the sine formula also, which is far too long-winded a method.

There are two ways of dealing with vectors … the good old trigonometry way that the ancient Greeks would have used, and the coordinate method.​

You've used the slow ancient Greek way.

Your professor wants you to use the quicker coordinate way.

Do what your professor suggested …

Thank you for the welcome!

I should have known I was doing it the long way, haha. I am VERY bad for going the more complicated routes because...well they seem easier...Yeah I'm odd.

But thank you for the help! I finally got it after I looked at it for bout another hour, haha.

Anyways, thanks again!