A 1000 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the kinetic frictional force f_k between the 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.
To find the time required for the boat to slow down to 45 km/h, we need to determine the acceleration of the boat first. We can then use the acceleration to find the time using the kinematic equation.
Let's start by converting the initial speed and final speed to meters per second:
Initial speed = 90 km/h = (90 * 1000) / 3600 m/s = 25 m/s
Final speed = 45 km/h = (45 * 1000) / 3600 m/s = 12.5 m/s
Since the boat's engine is shut off, the only force acting on the boat is the frictional force f_k. The frictional force f_k is given by f_k = 70v, where v is the speed of the boat in m/s.
Now, we need to determine the acceleration of the boat. The acceleration of an object can be calculated using the formula:
a = (v_f - v_i) / t
where a is the acceleration, v_f is the final velocity, v_i is the initial velocity, and t is the time.
Rearranging the equation, we have:
a = (12.5 m/s - 25 m/s) / t
Now, let's substitute the value of a into Newton's second law: F = ma
f_k = ma
We know that the mass of the boat (m) is 1000 kg, and the frictional force f_k is given by f_k = 70v.
Therefore,
70v = 1000 * a
Now, substitute the value of a from the previous equation into the acceleration equation:
70v = 1000 * ((12.5 m/s - 25 m/s) / t)
Simplifying the equation:
70v = 1000 * (-12.5 m/s) / t
Now, we can solve for t:
t = 1000 * (-12.5 m/s) / (70v)
Plugging in the given initial velocity (v = 25 m/s) and solving for t:
t = 1000 * (-12.5 m/s) / (70 * 25 m/s)
t = -35.71 seconds
The negative sign indicates that the acceleration is in the opposite direction of the motion. However, in this context, we are only interested in the magnitude of time. Therefore, we can ignore the negative sign.
So, the time required for the boat to slow down to 45 km/h is approximately 35.71 seconds.
To find the time required for the boat to slow to 45 km/h, we need to convert the given speeds into meters per second (m/s).
Given:
Mass of the boat (m): 1000 kg
Initial speed (v_initial): 90 km/h
Final speed (v_final): 45 km/h
Frictional force (f_k) = 70v
First, let's convert the initial and final speeds from km/h to m/s:
v_initial = 90 km/h * (1000 m/1 km) * (1 h/3600 s) = 25 m/s
v_final = 45 km/h * (1000 m/1 km) * (1 h/3600 s) = 12.5 m/s
Now, let's set up the equation to solve for time (t):
Initial speed - Final speed = Acceleration * Time
v_initial - v_final = (f_k/m) * t
Given the relationship between frictional force (f_k) and speed (v):
f_k = 70v
Substituting the value of f_k into the equation:
v_initial - v_final = (70v/m) * t
Now, substitute the given values into the equation:
25 m/s - 12.5 m/s = (70v/1000 kg) * t
12.5 m/s = (70 * 12.5/1000) * t
12.5 m/s = 0.875 t
Divide both sides by 0.875 to isolate t:
t = 12.5 m/s / 0.875
t ≈ 14.29 seconds
Therefore, the time required for the boat to slow to 45 km/h is approximately 14.29 seconds.
An exact solution will require calculus, since the acceleration is not constant.
M*dV/dt = -fk = -70V
dV/V = -(70/M)dt
Since you have separated variables to opposite sides, the differential equation is easily integrated.
ln V2/V1 = -ln2 = (-70/M)T
where T is the time interval.
T= (M/70)*ln2 = (1000/70)*0.693
= 99 seconds