A child is sitting on a stationary skateboard with a pile of rocks which she throws in one direction in order to make the skateboard travel in the opposite direction. If the rocks each have a mass of 0.70 kg and she can throw them with a speed of 13 m/s relative to the ground, determine the number of rocks she must throw per minute in order to maintain a constant average speed against a 2.0 N force of friction. (Note: because this is a rate, the answer may not be a whole number.)
To find the number of rocks the child must throw per minute to maintain a constant average speed against the force of friction, we need to consider the principles of Newton's laws of motion and the conservation of momentum.
First, let's calculate the initial momentum of one rock before it's thrown. The momentum (p) of an object is the product of its mass (m) and velocity (v):
p = m * v
Given that the mass of each rock is 0.70 kg and the speed at which the rocks are thrown is 13 m/s, the initial momentum of one rock can be calculated as:
p = 0.70 kg * 13 m/s = 9.10 kg*m/s
According to the law of conservation of momentum, the total momentum of the system (i.e., the child and the skateboard) must remain constant in the absence of external forces. Since the rocks are thrown in one direction, the child and the skateboard will move in the opposite direction to conserve momentum.
Now, let's consider the force of friction that opposes the motion of the skateboard. The force of friction (F) can be calculated using the equation:
F = μ * N
where μ is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the child and the skateboard, which we'll assume is 2.0 N.
Next, we can determine the acceleration (a) experienced by the child and the skateboard using Newton's second law:
F = m * a
Substituting the values, we have:
2.0 N = (m + rocks_mass) * a
Since in every minute, the child throws a number of rocks, we can calculate the mass of the rocks being thrown per minute (rocks_mass) using the following formula:
rocks_mass = (number_of_rocks * rock_mass) / time_in_min
where rock_mass is the mass of each rock (0.70 kg), number_of_rocks is the number of rocks thrown per minute, and time_in_min is the duration of one minute.
Finally, we solve the equation for a:
2.0 N = (m + rocks_mass) * a
a = 2.0 N / (m + rocks_mass)
Now, to maintain a constant average speed, the net force acting on the child and skateboard system should be zero. Therefore, the force of the rocks being thrown should counterbalance the force of friction. In other words, the force from the rocks should have the same magnitude as the force of friction:
2.0 N = number_of_rocks * p / t
where t is the time taken to throw each rock.
Combining the equations, we have:
2.0 N = number_of_rocks * p / t
Solving for the number_of_rocks:
number_of_rocks = (2.0 N * t) / p
Since the mass of each rock is 0.70 kg, and the speed at which the rocks are thrown is 13 m/s, we can substitute the known values to find the number_of_rocks per minute needed to maintain a constant average speed against a 2.0 N force of friction.