A 40.7kg wagon is towed up a hill inclined at 18.1 degrees with respect to the horizontal. The tow rope is parallel to the incline and has a tension of 139N in it. Assume that the wagon starts from rest at the bottom of the hill, and neglect friction. The acceleration of gravity is 9.8 m/s^2. How fast is the wagon going after moving 84.1m up the hill?
To find the speed of the wagon after moving up the hill, we can use the principles of work and energy.
First, we need to find the gravitational potential energy the wagon gains as it moves up the hill. The formula to calculate gravitational potential energy (PE) is:
PE = mgh
where m is the mass of the wagon (40.7 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height the wagon moves up the hill. To find h, we need to use trigonometry.
h = d * sin(theta)
where d is the distance the wagon moves up the hill (84.1 m) and theta is the angle of incline (18.1 degrees).
Now we can substitute the values into the equation to calculate the gravitational potential energy gained by the wagon:
PE = mgh = (40.7 kg) * (9.8 m/s^2) * (84.1 m * sin(18.1 degrees))
Next, we convert the gravitational potential energy into kinetic energy (KE) using the work-energy principle. The total amount of work done on the wagon is equal to the sum of the change in its kinetic energy and the change in its potential energy:
Work = KE + PE
Since the wagon starts from rest, its initial kinetic energy is zero. Therefore, the work done is equal to the change in potential energy:
Work = PE
Now we can equate the gravitational potential energy gained to the work done:
(40.7 kg) * (9.8 m/s^2) * (84.1 m * sin(18.1 degrees)) = Work
Finally, we can solve for the velocity (v) of the wagon at the top of the hill using the formula for work:
Work = F * d * cos(theta)
where F is the tension in the tow rope (139 N), d is the distance the wagon moves up the hill (84.1 m), and theta is the angle of incline (18.1 degrees).
(139 N) * (84.1 m) * cos(18.1 degrees) = Work
Once we find the value of Work, we can use the work-energy principle again to find the velocity of the wagon (v):
Work = KE
KE = (1/2) * m * v^2
Solve for v:
(1/2) * (40.7 kg) * v^2 = Work
Finally, we can calculate the speed (s) of the wagon by taking the square root of the velocity (v):
s = sqrt(v)
Substituting the values and solving these equations will give us the speed of the wagon after moving 84.1 m up the hill.