Jason takes off across level water on his jet-powered skis. The combined mass of Jason and skis is 75 kg (the mass of the fuel is negligible). The skis have a thrust of 200 N and a coefficient of kinetic friction on water of 0.1. Unfortunately, the skis run out of fuel after only 38 s. How far has Jason traveled when he finally coasts to a stop?
F = m a
during thrust portion
200 - mu m g = m a
200 = m (mu g + a)
200 = 75 (.981 + a)
a = 1.69 m/s^2
distance under power = d = (1/2) a t^2
= (1/2)1.69 (38)^2 = 1217 meters
by then is going
a t = 1.69*38 = 64.2 m/s
while deaccelerating with zero thrust
Vi = 64.2
F = -mu m g = m a
a = -mu g = -.981
0 = 64.2 - .981 t
t = 65.5 seconds
average speed = Vi/2 = 32.1 m/s
distance = 32.2 * 65.5 = 2107 meters more coasting
total distance = 1217+2107 = 3325 meters
Well, if Jason's skis run out of fuel, I suppose they'll be feeling a bit "jet lagged"! But let's get down to the calculations.
We can start by finding the initial acceleration using the formula:
F_net = m * a
Where F_net is the net force, m is the mass, and a is the acceleration. In this case, the net force is equal to the thrust minus the force of friction:
F_net = F_thrust - F_friction
F_friction = μ * m * g
Where μ is the coefficient of kinetic friction and g is the acceleration due to gravity. Plugging in the given values, we have:
F_friction = 0.1 * 75 kg * 9.8 m/s^2
Now, let's calculate the net force:
F_net = 200 N - (0.1 * 75 kg * 9.8 m/s^2)
Next, we can find the initial acceleration:
F_net = m * a
a = F_net / m
Finally, we can use the kinematic equation:
v = u + at
Where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time.
Since v is the final velocity, which is also 0 when Jason coasts to a stop, we can rearrange the equation to solve for t:
t = -u / a
Plugging in the values, we can calculate the time it takes for Jason to coast to a stop.
To find the distance Jason has traveled when he finally stops, we need to determine the acceleration and time it takes for him to stop.
First, let's calculate the force of friction acting on Jason and his skis. The frictional force can be found using the equation:
frictional force = coefficient of kinetic friction * normal force
The normal force is equal to the weight of Jason and his skis, which can be calculated as:
weight = mass * acceleration due to gravity
Given that the mass of Jason and his skis is 75 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate:
weight = 75 kg * 9.8 m/s^2 = 735 N
Using a coefficient of kinetic friction of 0.1, we can find the frictional force:
frictional force = 0.1 * 735 N = 73.5 N
The net force acting on Jason and his skis can be calculated as the difference between the thrust and the frictional force:
net force = thrust - frictional force = 200 N - 73.5 N = 126.5 N
Next, we can calculate the acceleration:
acceleration = net force / mass = 126.5 N / 75 kg = 1.6867 m/s^2
Since we know that the skis run out of fuel after 38 seconds, we can calculate the initial velocity using the equation of motion:
v = u + at
where:
v = final velocity (0 m/s, since Jason eventually stops)
u = initial velocity
a = acceleration
t = time
Plugging in the values, we have:
0 = u + (1.6867 m/s^2)(38 s)
Rearranging the equation, we find:
u = -1.6867 m/s^2 * 38 s = -64.0 m/s
(Note: The negative sign indicates deceleration.)
Now, we can calculate the distance traveled using the equation of motion:
s = ut + (1/2)at^2
Given that u = -64.0 m/s, a = 1.6867 m/s^2, and t = 38 s, we have:
s = (-64.0 m/s)(38 s) + (1/2)(1.6867 m/s^2)(38 s)^2
Calculating this equation will give us the distance traveled by Jason before stopping.
To find out how far Jason has traveled when he finally coasts to a stop, we need to calculate the total distance covered.
Let's break down the problem step by step using Newton's second law and the equations of motion.
First, let's find out the net force acting on Jason and his skis. The net force can be calculated as the difference between the thrust force and the frictional force:
Net Force = Thrust Force - Frictional Force
The thrust force is given as 200 N. To calculate the frictional force, we need to multiply the coefficient of kinetic friction by the normal force. The normal force is equal to the weight of Jason and the skis, which can be calculated by multiplying the mass by the acceleration due to gravity (9.8 m/s^2):
Normal Force = Mass * Acceleration due to gravity
Normal Force = 75 kg * 9.8 m/s^2 = 735 N
Frictional Force = Coefficient of Kinetic Friction * Normal Force
Frictional Force = 0.1 * 735 N = 73.5 N
Net Force = Thrust Force - Frictional Force
Net Force = 200 N - 73.5 N
Net Force = 126.5 N
Now, let's calculate the acceleration using Newton's second law:
Net Force = Mass * Acceleration
Acceleration = Net Force / Mass
Acceleration = 126.5 N / 75 kg
Acceleration = 1.687 m/s^2
Since the skis run out of fuel after 38 seconds, we can calculate the initial velocity using the equation of motion:
Initial Velocity = Acceleration * Time
Initial Velocity = 1.687 m/s^2 * 38 s
Initial Velocity = 64.106 m/s
Next, let's calculate the distance traveled using the equation of motion:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Distance = 64.106 m/s * 38 s + (1/2) * 1.687 m/s^2 * (38 s)^2
Distance = 2431.428 m + 1/2 * 1.687 m/s^2 * 1444 s^2
Distance = 2431.428 m + 1219.056 m
Distance = 3650.484 m
Therefore, Jason will have traveled approximately 3650.484 meters (or 3.65 kilometers) when he finally coasts to a stop.