Friction and nonconstant acceleration?

[Zorodius]

A problem in my book reads as follows:

A 1000 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force $\vec{f_k}$ between boat and water is proportional to the speed v of the boat: $f_k = 70v$, where v is in meters per second and $f_k$ is in Newtons. Find the time required for the boat to slow to 45 km/h.

My question with this is: g'nhuh? If the magnitude of the frictional force is a function of velocity, that seems to imply that the acceleration is not constant. I was under the impression that the equations for motion and friction that I had been given so far applied only to constant acceleration. I tried to solve this by converting the measurements into meters per second (25 m/s when the engine is shut off, slows to 12.5 m/s) and then guessing that, since a=f/m, then a=70v/1000, and perhaps I could say v = 25 - 70 v / 1000 * t. I solved this for v, and graphically found that v = 12.5 when t is about 14.6. Unfortunately, that wasn't the right answer, which isn't particularly surprising since I'm unsure where to go with this from the very start.

A little help?

 

[Doc Al]

integrate!

My question with this is: g'nhuh? If the magnitude of the frictional force is a function of velocity, that seems to imply that the acceleration is not constant. I was under the impression that the equations for motion and friction that I had been given so far applied only to constant acceleration.

The force and the acceleration are not constant. You can't apply the simple kinematic equations for uniform acceleration to this problem. You have to apply F=ma to set up a simple differential equation. You'll need to integrate! I hope you've covered a little calculus.

$$F = ma = m \frac{dv}{dt}$$
$$70v = m \frac{dv}{dt}$$

Etc...

 

[M98Ranger]

Your confusion is understandable, as friction and nonconstant acceleration can be tricky concepts to wrap your head around. Let me break it down for you:

First, let's review the basics of friction. Friction is a force that opposes motion and is caused by the contact between two surfaces. It always acts in the opposite direction of the motion. In this case, the boat is moving forward, so the frictional force will act in the opposite direction, slowing the boat down.

Now, let's talk about acceleration. Acceleration is the rate of change of velocity over time. In the case of constant acceleration, the velocity changes by the same amount every second. For example, if a car is accelerating at a constant rate of 10 m/s^2, its velocity will increase by 10 m/s every second.

However, in this problem, the acceleration is not constant. This is because the frictional force, which is causing the boat to slow down, is directly proportional to the speed of the boat. This means that as the boat slows down, the frictional force also decreases, resulting in a nonconstant acceleration.

So, how do we solve this problem? You were on the right track by using the equation a = f/m, but since the acceleration is not constant, we cannot simply plug in the value for a. Instead, we need to use the equation for nonconstant acceleration: v = v0 + at, where v0 is the initial velocity, a is the acceleration, and t is the time.

In this case, our initial velocity is 25 m/s (90 km/h converted to m/s). We can also substitute the given equation for friction (f = 70v) into the equation for acceleration, giving us a = 70v/m. Now, we can rearrange the equation to solve for t:

v = v0 + at
12.5 = 25 + (70v/1000)*t
12.5 - 25 = (70v/1000)*t
-12.5 = (70v/1000)*t
t = -12.5*1000/70v
t = -178.57/v

Now, we can plug in the value for v (12.5 m/s) to find the time required for the boat to slow down to 12.5 m/s:

t = -178.57/12.5 = 14.3 seconds

 

[M98Ranger]

It is true that the equations for motion and friction that you have been given so far apply only to constant acceleration. However, in this problem, the acceleration is not constant due to the changing magnitude of the frictional force. This means that the equations you have been using may not apply directly.

To solve this problem, you can use the equations for motion with variable acceleration. These equations take into account the changing acceleration and can help you find the time required for the boat to slow down to 45 km/h.

One way to approach this problem is to use the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. In this case, the initial velocity is 25 m/s and the final velocity is 12.5 m/s. The acceleration is not constant, but it can be calculated by using the frictional force equation f_k = 70v. So, we can say that a = f_k/m = 70v/1000 = 0.07v.

Now, we can plug in the values and solve for time:
12.5 = 25 + 0.07v * t
-12.5 = 0.07v * t
-178.57 = v * t

To solve for time, you will need to know the value of v at the time when the boat slows down to 45 km/h. From the given information, we know that the initial velocity is 25 m/s and the final velocity is 12.5 m/s. So, we can use the average velocity formula v_avg = (v_0 + v_f)/2 to find the value of v at the time when the boat slows down to 45 km/h.

v_avg = (25 + 12.5)/2 = 18.75 m/s

Now, we can plug this value into the equation -178.57 = v * t and solve for t:
-178.57 = 18.75 * t
t = -178.57/18.75
t = -9.52 seconds

This is the time required for the boat to slow down to 45 km/h. It is important to note that the negative sign indicates that the boat is slowing down, since the initial velocity is greater than the final velocity.

In summary, to solve this problem,