A 1250 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force fk between boat and water is proportional to the speed v of the boat. Thus, fk = 80v, where v is in meters per second and fk (the magnitude of the frictional force) is in newtons. Find the time required for the boat to slow down to 45 km/h.
First, we need to convert the initial speed and the desired final speed into meters per second.
Initial speed = 90 km/h
To convert km/h to m/s, we divide by 3.6:
90 km/h ÷ 3.6 = 25 m/s
Final speed = 45 km/h
To convert km/h to m/s, we divide by 3.6:
45 km/h ÷ 3.6 = 12.5 m/s
Now, let's find the initial frictional force (fk) when the engine is shut off.
Given: fk = 80v, where v is the speed of the boat in m/s.
Initial speed v = 25 m/s
fk = 80 * 25 = 2000 N
The net force acting on the boat is the difference between the initial frictional force (fk) and the force due to inertia (F = ma, where m is mass and a is acceleration). The acceleration can be calculated using the formula a = F/m.
Net force = fk - F
The force due to inertia, F = ma
Since the engine is shut off, there is no external force acting on the boat. So, the net force is the frictional force (fk) opposing the motion.
Net force = fk
Using the formula a = F/m, we can solve for acceleration:
a = fk / m
a = 2000 N / 1250 kg = 1.6 m/s^2
The acceleration is constant, so we can use the equation of motion: v = u + at, where
v = final velocity
u = initial velocity
a = acceleration
t = time taken
Substituting the given values:
12.5 m/s = 25 m/s + (1.6 m/s^2)t
Rearranging the equation to solve for t:
12.5 m/s - 25 m/s = (1.6 m/s^2)t
-12.5 m/s = (1.6 m/s^2)t
t = -12.5 m/s / (1.6 m/s^2)
t ≈ -7.81 s
The negative time value indicates that the boat has already slowed down to 12.5 m/s. However, we are interested in the time it takes for the boat to slow down to 12.5 m/s. To find the positive time value, we take the absolute value:
t ≈ | -7.81 s | ≈ 7.81 s
Therefore, it would take approximately 7.81 seconds for the boat to slow down to 45 km/h.
To solve for the time required for the boat to slow down to 45 km/h, we need to find the change in velocity and then divide it by the acceleration.
First, let's convert the initial velocity and the desired final velocity to meters per second (m/s).
Initial velocity (v1) = 90 km/h
Final velocity (v2) = 45 km/h
1 km/h = 0.2778 m/s, so we can convert the velocities as follows:
v1 = 90 km/h * 0.2778 m/s = 25 m/s
v2 = 45 km/h * 0.2778 m/s = 12.5 m/s
Now, let's calculate the change in velocity (Δv):
Δv = v2 - v1 = 12.5 m/s - 25 m/s = -12.5 m/s
Since the velocity is decreasing, we use a negative sign to indicate the direction.
Next, let's find the acceleration (a) using the frictional force equation:
fk = 80v (given)
At this point, we need to determine the velocity (v) that corresponds to the frictional force (fk) at each velocity.
Substitute the given velocity (v1) into the equation:
fk = 80 * v1 = 80 * 25 = 2000 N
Now, substitute the desired velocity (v2) into the equation:
fk = 80 * v2 = 80 * 12.5 = 1000 N
Therefore, we can see that the acceleration is proportional to the change in velocity:
a = (1000 N - 2000 N) / Δv = -1000 N / -12.5 m/s = 80 m/s^2
Finally, to find the time required for the boat to slow down to 45 km/h, we can use the following equation:
Δv = a * t
Rearranging the equation, we can solve for time (t):
t = Δv / a = (-12.5 m/s) / (80 m/s^2) ≈ -0.15625 s
Since time cannot be negative in this context, we take the absolute value of the time to get the positive value:
t ≈ | -0.15625 s | = 0.15625 seconds
Therefore, the time required for the boat to slow down to 45 km/h is approximately 0.15625 seconds.