Sam, whose mass is 82 {\rm kg}, takes off across level snow on his jet-powered skis. The skis have a thrust of 240 {\rm N} and a coefficient of kinetic friction on snow of 0.11. Unfortunately, the skis run out of fuel after only 15 {\rm s}.
To find out how far Sam travels before running out of fuel, we can use the concept of work and energy. The work done by the thrust provided by the skis will equal the change in Sam's kinetic energy.
1. First, we need to calculate the net force acting on Sam.
- The thrust force provided by the skis is 240 N.
- The force of kinetic friction can be calculated using the formula: frictional force = coefficient of kinetic friction * normal force.
- The normal force on Sam is equal to his weight, which can be calculated using the formula: weight = mass * gravitational acceleration, where the gravitational acceleration is approximately 9.8 m/s².
Therefore, the frictional force is given by: frictional force = 0.11 * (82 kg * 9.8 m/s²).
2. Next, we need to calculate the net force acting on Sam.
- The net force is given by: net force = thrust force - frictional force.
3. To find out the acceleration of Sam, we can use Newton's second law: net force = mass * acceleration.
- Rearranging the formula, we can calculate the acceleration: acceleration = net force / mass.
4. Now that we know the acceleration, we can calculate the distance traveled by Sam using the formula: distance = initial velocity * time + (1/2) * acceleration * time².
- Since Sam starts from rest, the initial velocity is 0 m/s.
Therefore, the total distance traveled by Sam before running out of fuel is given by the distance equation above.
Remember to convert the mass from kg to kg, force from N to N, and time from s to s to ensure consistency throughout the calculations.
To find the acceleration of Sam's jet-powered skis, we can use the equation:
net force = mass × acceleration
The net force acting on the skis can be calculated as the difference between the force provided by the jet engine and the force of friction. The force of friction can be calculated as the product of the coefficient of kinetic friction and the normal force. In this case, the normal force is equal to the weight of Sam, which can be calculated as the product of his mass and the acceleration due to gravity.
Given:
Mass of Sam (m) = 82 kg
Thrust of skis (F_thrust) = 240 N
Coefficient of kinetic friction (μ) = 0.11
Time taken (t) = 15 s
Step 1: Calculate the weight of Sam
Weight (W) = mass × acceleration due to gravity
W = 82 kg × 9.8 m/s^2
W ≈ 803.6 N
Step 2: Calculate the force of friction
Force of friction (F_friction) = coefficient of kinetic friction × normal force
F_friction = 0.11 × 803.6 N
F_friction ≈ 88.4 N
Step 3: Calculate the net force
Net force (F_net) = F_thrust - F_friction
F_net = 240 N - 88.4 N
F_net ≈ 151.6 N
Step 4: Calculate the acceleration
F_net = m × a
a = F_net / m
a ≈ 151.6 N / 82 kg
a ≈ 1.85 m/s^2
Therefore, the acceleration of Sam's jet-powered skis is approximately 1.85 m/s^2.