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.
To find the time required for the boat to slow down to 45 km/h, we need to determine the initial and final speeds and use the given frictional force equation.
Given:
- Mass of the boat (m) = 1250 kg
- Initial speed (v₁) = 90 km/h
- Final speed (v₂) = 45 km/h
- Frictional force (fk) = 80v, where v is in m/s in the equation.
First, let's convert the given speeds from km/h to m/s.
1 km/h is equal to 0.2778 m/s.
Initial speed (v₁) = 90 km/h * (0.2778 m/s / 1 km/h) = 25 m/s
Final speed (v₂) = 45 km/h * (0.2778 m/s / 1 km/h) = 12.5 m/s
Now, we can solve for the time required (t) using the equation of motion:
fk = (m * Δv) / t
In this case, Δv (change in velocity) is equal to the difference between the initial and final speeds: Δv = v₂ - v₁.
Since the magnitude of the frictional force "fk" is proportional to the speed "v," we can write the equation as:
fk = 80v
By substituting the equation for the frictional force and rearranging, we get:
80v = (m * Δv) / t
t = (m * Δv) / (80v)
Now, let's plug in the given values:
m = 1250 kg
Δv = v₂ - v₁ = 12.5 m/s - 25 m/s = -12.5 m/s (negative sign indicates deceleration)
v = v₁ = 25 m/s
t = (1250 kg * (-12.5 m/s)) / (80 * 25 m/s)
Simplifying further:
t = (-15625 kg * m/s) / (2000 m/s)
t = -7.81 seconds
Note: The negative sign in the value of time means that this is the time taken for the boat to decelerate from 90 km/h to 45 km/h.
Therefore, the time required for the boat to slow down to 45 km/h is approximately 7.81 seconds.