A child pushes a wagon with a passenger of total mass 42 kg along a horizontal surface as fast as the
child can run, and then releases the wagon, which continues for another 16 m before stopping. The coefficient
of kinetic friction acting to slow the wagon is 0.18. What was the speed of the boy when he released the wagon?
work (energy) to stop the wagon
... w = 42 * g * .18 * 16
w equals the initial KE of the wagon
... 1/2 * 42 * v^2 = w
the speed of the boy is the same as the initial speed of the wagon
To find the speed of the boy when he released the wagon, we can use the concept of conservation of mechanical energy.
1. First, let's find the work done by the boy in pushing the wagon. The work done is equal to the change in kinetic energy.
Work done = Change in kinetic energy
2. The initial kinetic energy of the wagon is zero since it starts from rest.
3. The final kinetic energy of the wagon can be calculated using the equation:
Final kinetic energy = (1/2) * mass * velocity^2
4. The work done by the boy pushing the wagon is equal to the change in kinetic energy, which is equal to the final kinetic energy of the wagon.
Work done = (1/2) * mass * velocity^2
5. The work done by the boy is done against the force of friction. The work done against friction is given by the equation:
Work done against friction = force of friction * distance
The force of friction can be calculated using the equation:
Force of friction = coefficient of kinetic friction * normal force
6. The normal force is equal to the weight of the wagon, which is given by:
Normal force = mass * acceleration due to gravity
7. Substitute the value of the force of friction and distance into the equation to find the work done against friction.
Work done against friction = (coefficient of kinetic friction * normal force) * distance
8. Equate the work done against friction to the work done by the boy.
(coefficient of kinetic friction * normal force) * distance = (1/2) * mass * velocity^2
9. Rearrange the equation to solve for the velocity:
velocity = sqrt((2 * (coefficient of kinetic friction * normal force) * distance) / mass)
10. Substitute the given values into the equation and calculate the velocity.
mass = 42 kg, coefficient of kinetic friction = 0.18, distance = 16 m, acceleration due to gravity = 9.8 m/s^2
The final velocity is the speed of the boy when he released the wagon.
To determine the speed of the boy when he released the wagon, we need to use the law of conservation of energy. The initial kinetic energy of the wagon and the passenger, when pushed by the boy, will be converted into work done against the friction and the potential energy when the wagon comes to a stop.
Here are the steps to calculate the speed of the boy:
1. Calculate the work done against friction:
The work done against friction is given by the equation:
Work = Force × Distance × cos(θ)
Since the wagon is moving horizontally and the force of friction opposes the motion, the angle between the force and distance is 180 degrees.
So, Work = Force of Friction × Distance
Work = μ × Normal Force × Distance, where μ is the coefficient of kinetic friction.
2. Calculate the normal force:
The normal force is the force exerted by the surface on the wagon. Since the wagon is on a horizontal surface, the normal force is equal to the weight of the wagon and the passenger.
Normal Force = mass × gravity
3. Calculate the work-energy theorem:
The work done against friction is equal to the change in kinetic energy.
Work = ΔKE
ΔKE = 0.5 × mass × (final velocity^2 - initial velocity^2)
Since the wagon comes to a stop, the final velocity is 0.
4. Rearrange the equation to solve for the initial velocity:
0.5 × mass × (initial velocity^2) = μ × Normal Force × Distance
5. Substitute the given values into the equation and solve for the initial velocity:
mass = 42 kg
μ = 0.18
Distance = 16 m
gravity = 9.8 m/s^2
Plug these values into the equation and solve for the initial velocity of the boy.
Using these steps, you can calculate the speed of the boy when he released the wagon.