When a 644 lb boat is moving at 10fps, the motor conks out. How much farther will the boat
glide, assuming its resistance to motion is 2v lb where v is in fps.
Well, if the boat is moving at 10 feet per second and the motor conks out, I guess you could say it's going to be 'boat' a lot of trouble! But let's put the laughter aside for a moment and do some calculations.
We can use Newton's second law to find the deceleration of the boat when the motor conks out. The resistance to motion is given by 2v lb, where v is in fps. So when the boat is moving at 10fps, the resistance is 2(10) = 20 lb.
The deceleration of the boat can be calculated using the equation F = ma, where F is the net force acting on the boat, m is the mass, and a is the acceleration. The net force is given by F = ma = 20 lb.
Now we need to find the deceleration. Rearranging the equation F = ma, we get a = F/m = 20 lb/644 lb ≈ 0.031 fps².
Since the boat is decelerating, we need to find the time it takes for the boat to come to a stop. We can 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, v is 0 fps, u is 10 fps, and a is -0.031 fps² (negative because it's decelerating).
Rearranging the equation, we get t = (v - u) / a = (0 - 10) / (-0.031) ≈ 322.58 seconds.
Finally, we can find the distance the boat will glide by using the equation s = ut + 0.5at², where s is the distance, u is the initial velocity, t is the time, and a is the acceleration.
Plugging in the values, we get s = (10)(322.58) + 0.5(-0.031)(322.58)² ≈ 3225.8 feet.
So, the boat will glide approximately 3225.8 feet before coming to a stop. That's quite a 'shore' bet!
To find out how much farther the boat will glide, we need to calculate the distance it will travel before coming to a complete stop.
Given:
Mass of the boat (m) = 644 lb
Initial velocity (u) = 10 fps
Resistance to motion (R) = 2v lb, where v is in fps
The force of resistance to motion (Fr) is given by the formula:
Fr = m * R
Since R = 2v, we can rewrite the equation as:
Fr = m * 2v
The force acting against the motion reduces the boat's velocity according to Newton's second law of motion:
Fr = ma
Where a is the acceleration. Rearranging the equation, we have:
a = Fr / m
The acceleration is also the rate of change of velocity (v), so:
a = dv / dt
Rearranging the equation, we have:
dv = a * dt
Integrating both sides, we get:
∫dv = ∫a * dt
v = ∫a * dt
To find the integral of a with respect to t, we need to substitute a with Fr / m:
v = ∫(Fr / m) * dt
v = (1 / m) * ∫Fr * dt
v = (1 / m) * ∫(m * 2v) * dt
v = 2 ∫v * dt
v = 2vt + C
Since the initial velocity u = 10 fps:
u = 2(10)t + C
At t = 0, u = 10, so we can find C:
10 = 0 + C
C = 10
Now we have the equation for v:
v = 2vt + 10
Rearranging the equation:
dt = dv / (2vt + 10)
Integrating both sides:
∫dt = ∫dv / (2vt + 10)
t = (1/2) * ln|2vt + 10| + K
Where K is the constant of integration. At t = 0, v = 10, so we can find K:
0 = (1/2) * ln|2(10)(0) + 10| + K
0 = (1/2) * ln|0 + 10| + K
0 = (1/2) * ln(10) + K
K = - (1/2) * ln(10)
Now we have the equation for t:
t = (1/2) * ln|2vt + 10| - (1/2) * ln(10)
Since we want to find the distance traveled, we use the formula:
d = ut + (1/2)at^2
The acceleration a is equal to the force of resistance to motion divided by the mass (Fr / m):
a = Fr / m
Let's substitute the values and calculate the distance traveled by the boat before coming to a stop.
To determine how much farther the boat will glide when the motor conks out, we need to calculate the distance it travels before coming to a stop due to resistance.
In this scenario:
- The boat weighs 644 lb.
- It is moving at a speed of 10 fps.
- The resistance to motion is given by 2v lb, where v is in fps.
We can break down the problem into steps:
Step 1: Calculate the deceleration due to resistance
The deceleration is given by the force of resistance divided by the mass of the boat.
Force of resistance = 2v lb
Mass of the boat = 644 lb
Deceleration = Force of resistance / Mass of the boat
Deceleration = (2v lb) / (644 lb)
Step 2: Calculate the time taken to come to a stop
The time taken to come to a stop can be found by dividing the initial velocity by the deceleration.
Time = Initial velocity / Deceleration
Time = 10 fps / ((2v lb) / (644 lb))
Step 3: Calculate the distance traveled during this time
The distance traveled can be found using the equation:
Distance = (Initial velocity * Time) - (0.5 * (Deceleration) * (Time^2))
Distance = (10 fps * Time) - (0.5 * (Deceleration) * (Time^2))
Now, you can plug in the value of Time obtained from Step 2 and simplify the equation to find the distance the boat glides before coming to a stop.
Please note that the answer will be in feet since we are using feet per second (fps) for velocity.
Oh forgot you want x
x = -100 e^-t/10 + c
if c = 100
then x = 0 at 0 and 100 at t = infinity
Well, the last time I did this in pounds and slugs was about 1956 but
in your units g = 32 ft/s^2 approximately
m = mass of boat in slugs = weight / g = 644/32 = 20.1 slugs
F = m a
2 v = -20.1 a = -20.1 dv/dt
2 v + 20.1 dv/dt = 0
let v = A e^kt
then dv/dt = k A e^kt = k v
2 v + 20.1 k v = 0
k = -2/20.1 = - 1/10 approx
v = Vi e^-t/10 = 10 e^-t/10
well it goes slower and slower but never hits zero
let's look at 10 seconds
v at 10 = 10 e^-1 = 3.7
look at 60 seconds
v at 60 = 10 e^-6 = 0.025 well that is pretty near zero :)