What Speed Would the Arrow Leave the Bow if Force Were Doubled?

[Amber430]

This problem really shouldn't be this difficult but for some reason I can't seem to find the answer.

An arrow, starting from rest, leaves the bow with a speed of 26.0 m/s. If the average force exerted on the arrow by the bow were doubled, all else remaining the same, with what speed would the arrow leave the bow?

net force= mass x acceleration

Acceleration is unknown in this question. Velocity (I'm assuming it's the final V?) is known (26.0 m/s). Initial velocity is 0. I know that when force doubles, acceleration doubles. That's about as far as I could get.

 

[Doc Al]

I know that when force doubles, acceleration doubles.

Good. What quantity remains the same? When they said "all else remaining the same", what could they have been talking about?

 

[Amber430]

The initial velocity remains the same. But the only other variable that is given is final velocity, which would not stay the same.

 

[Doc Al]

Hint: The distance the arrow takes to reach the final speed is the same.

How do distance, acceleration, and speed relate?

 

[Amber430]

V^2=Vo^2 + 2ax. But what is x? It isn't known.

 

[Doc Al]

All you want to do is compare the final speeds. You don't need to know the actual distance. Just call it x.

 

[Amber430]

But what is the acceleration?

 

[Doc Al]

But what is the acceleration?

Again, you don't need actual numbers. Call the original acceleration "a". What would be the new acceleration?

 

[Amber430]

It would be 2a

 

[Doc Al]

It would be 2a

Good. Now solve for the speeds in terms of "a" and "x" and compare. (You can also use ratios.)