Six dogs pull a two-person sled with a total mass of 280 kg. The coefficient of kinetic friction between the sled and the snow is 0.080. The sled accelerates at 0.65 m/s2 until it reaches a cruising speed of 15 km/h. What is the team's maximum power output during the acceleration phase? What is the team's power output during the cruising phase?
To find the team's maximum power output during the acceleration phase, we need to calculate the net force and then multiply it by the acceleration.
Step 1: Calculate the net force during the acceleration phase.
The net force can be determined using Newton's second law of motion:
Net Force = Mass * Acceleration
Given that the total mass of the sled and dogs is 280 kg and the acceleration is 0.65 m/s^2, we have:
Net Force = 280 kg * 0.65 m/s^2
Net Force = 182 N (Newtons)
Step 2: Calculate the maximum power output during the acceleration phase.
Power is defined as the rate at which work is done, and work is equal to force multiplied by distance:
Power = Work / Time
During the acceleration phase, the force applied by the dogs is overcoming the force of kinetic friction. The work done against friction can be given by:
Work = Force * Distance
Distance = (Cruising Speed)^2 / (2 * Acceleration)
Distance = (15 km/h)^2 / (2 * 0.65 m/s^2) [Converting 15 km/h to m/s: 15 km/h * (1000 m/km) / (3600 s/h) = 4.17 m/s]
Distance ≈ 26.8 m
Thus, the work done against friction is:
Work = (182 N) * (26.8 m)
Work = 4889.6 J (Joules)
Now, we need to determine the time it takes for the sled to reach its cruising speed. To do this, we can use the equation of motion:
Final Velocity = Initial Velocity + (Acceleration * Time)
Since the sled is starting from rest (initial velocity is 0), we can rearrange the equation and solve for time:
Time = Final Velocity / Acceleration
Time = 4.17 m/s / 0.65 m/s^2
Time ≈ 6.42 s
Therefore, the power output during the acceleration phase is:
Power = Work / Time
Power = 4889.6 J / 6.42 s
Power ≈ 761 W (Watts)
Now let's find the team's power output during the cruising phase.
Step 1: Calculate the net force during the cruising phase.
During the cruising phase, the sled is moving at a constant speed. This means the net force is zero since there is no acceleration.
Net Force = 0 N
Step 2: Calculate the power output during the cruising phase.
Since the net force is zero, no work is being done to maintain the speed. Thus, the power output during the cruising phase is also zero.
Power = 0 W
Therefore, the team's power output during the cruising phase is zero.
To find the team's maximum power output during the acceleration phase, we need to first calculate the net force acting on the sled and then multiply it by the sled's acceleration.
1. Calculate the net force:
The net force can be determined by subtracting the force of kinetic friction from the force of the dogs pulling the sled.
Force of dogs pulling the sled:
The force exerted by the dogs pulling the sled can be calculated using Newton's second law: F = ma, where F is the force, m is the total mass of the sled, and a is the acceleration.
F = (280 kg) × (0.65 m/s²)
F = 182 N
Force of kinetic friction:
The force of kinetic friction can be calculated using the formula: fk = μk × N, where fk is the force of kinetic friction, μk is the coefficient of kinetic friction, and N is the normal force.
The normal force is equal to the weight of the sled, which can be calculated as: N = mg, where m is the total mass and g is the acceleration due to gravity.
N = (280 kg) × (9.8 m/s²)
N = 2744 N
fk = (0.080) × (2744 N)
fk = 219.52 N
Net force = Force of dogs pulling the sled - Force of kinetic friction
Net force = 182 N - 219.52 N
Net force = -37.52 N (Since the force of kinetic friction and the force of the dogs pulling the sled are in opposite directions, we have a negative value for the net force.)
2. Calculate the power output during the acceleration phase:
Power is defined as the rate at which work is done, which can be calculated as the product of force and velocity: P = F × v, where P is power, F is force, and v is velocity.
Since power is the rate of doing work, we need to multiply the net force (due to friction) by the velocity of the sled during acceleration to get the sled's power output.
Power during acceleration phase = Net force × acceleration × v
The net force is the negative value calculated as -37.52 N (opposing the motion of the sled).
The acceleration is given as 0.65 m/s².
The velocity can be converted from km/h to m/s: v = (15 km/h) × (1000 m/1 km) × (1 h/3600 s)
Power during acceleration phase = (-37.52 N) × (0.65 m/s²) × (15,000 m/3600 s)
To find the team's power output during the cruising phase, we need to know the resistance force acting against the sled when it reaches a constant velocity.
3. Calculate the power output during the cruising phase:
At constant velocity, the net force is zero since there is no acceleration. Therefore, the force of the dogs pulling the sled is equal in magnitude and opposite in direction to the force of kinetic friction.
Power during cruising phase = force of dogs pulling the sled × velocity
Now we can calculate the power output during the acceleration and cruising phases.