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?
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To determine how far Jason has traveled when he finally coasts to a stop, we need to calculate the distance covered during two phases: when the skis are powered by the thrust and when they are coasting due to the kinetic friction on water.
Let's start with the first phase when the skis are powered by the thrust. The equation we'll use is:
F = m * a
Where:
F is the net force,
m is the mass of Jason and skis, and
a is the acceleration.
Since the skis have a thrust of 200 N and the mass is given as 75 kg, we can rearrange the equation to solve for acceleration:
a = F / m
a = 200 N / 75 kg
a ≈ 2.67 m/s²
Now, we can use the equations of linear motion to calculate the distance covered during this phase.
The equation to calculate the distance covered when the skis are powered by the thrust is:
d = v * t + 0.5 * a * t²
Where:
d is the distance covered,
v is the initial velocity (which is assumed to be 0 since the skis start from rest),
t is the time, and
a is the acceleration.
Plugging in the values:
d1 = 0.5 * a * t²
d1 = 0.5 * 2.67 m/s² * (38 s)²
d1 ≈ 2,411.97 m
This gives us the distance covered during the first phase when the skis are powered by the thrust.
Now, let's calculate the distance covered during the second phase when the skis are coasting due to the kinetic friction on water. The force of friction can be calculated as:
f_friction = μ * m * g
Where:
f_friction is the force of friction,
μ is the coefficient of kinetic friction on water (which is given as 0.1),
m is the mass of Jason and skis, and
g is the acceleration due to gravity.
f_friction = 0.1 * 75 kg * 9.8 m/s²
f_friction ≈ 73.5 N
Now, we can use the equation F = m * a to calculate the deceleration during this phase:
f_friction = m * a
73.5 N = 75 kg * a
a ≈ 0.98 m/s²
Using the equation of linear motion again, the distance covered during this phase can be calculated as:
d2 = v² / (2 * a)
Since the skis are coasting to a stop, the initial velocity (v) is the final velocity at the end of the first phase, which can be calculated using the equation v = a * t:
v = 2.67 m/s² * 38 s
v ≈ 101.46 m/s
Plugging in the values:
d2 = (101.46 m/s)² / (2 * 0.98 m/s²)
d2 ≈ 2617.57 m
Now, we can calculate the total distance covered by adding the distances covered during both phases:
total distance = d1 + d2
total distance ≈ 2,411.97 m + 2617.57 m
total distance ≈ 5,029.54 m
Therefore, Jason travels approximately 5,029.54 meters when he finally coasts to a stop.