Mechanics Problem Involving Newton's Laws

[OpticalFreeze]

Homework Statement

An arrow, starting from rest, leaves the bow with a speed of 19.5 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?

Homework Equations

f=ma

The Attempt at a Solution

I know that when the force is doubled, so is the acceleration and therefore the rate of change in velocity is increased. I know that the velocity is not simply doubled as the acceleration increases the velocity over time. I'm not sure where to go with this problem because every possible solution I can think of would involve me knowing the time or the displacement, which I can't figure out how to get from the problem.

Any help would be greatly appreciated.

 

[gabbagabbahey]

I think that the key phrase in the question is "all else being the same" So if the acceleration before doubling was calculated using a time interval T , you would use the same interval T in calculating the acceleration after the force is doubled. If T stays the same and a is doubled, what must happen to v?

 

[baba89]

I would approach this problem by first identifying the relevant equations and principles involved. In this case, we are dealing with Newton's laws of motion, specifically the second law which states that the net force on an object is equal to its mass multiplied by its acceleration (f=ma).

Next, I would draw a free body diagram of the arrow at two different points in time - when it is leaving the bow and when it has reached its maximum speed. This will help me visualize the forces acting on the arrow and how they change with the doubling of the force from the bow.

Using the equation f=ma, we can set up an equation for the initial and final velocities of the arrow:

Initial velocity: f1 = ma1 = 19.5 m/s
Final velocity: f2 = 2ma2 = unknown

We can then use the fact that the mass of the arrow remains constant to set up a ratio between the two equations:

f1/f2 = a1/a2

Substituting in the known values, we get:

19.5/f2 = a1/a2

Since we know that the acceleration is directly proportional to the force, we can write:

a1/a2 = f1/f2

Substituting in the known values again, we get:

19.5/f2 = 1/2

Solving for f2, we get:

f2 = 2*19.5 = 39 m/s

Therefore, if the force exerted on the arrow by the bow is doubled, the arrow will leave the bow with a speed of 39 m/s.

In conclusion, by applying Newton's laws of motion and using a systematic approach, we were able to solve this mechanics problem involving the doubling of force and its effect on an object's speed. It is important to always start with the relevant principles and equations, and to visualize the problem in order to find a solution.