A 40.0 kg wagon is towed up a hill inclined at 17.9° with respect to the horizontal. The tow rope is parallel to the incline and has a tension of 140 N. Assume that the wagon starts from rest at the bottom of the hill, and neglect friction. How fast is the wagon going after moving 75 m up the hill?
The central thing is to find the net force, up, then the acceleration, then the distance.
net force up=140-40g*sinTheta
acceleration= net force up/mass
Work done on gravity=mg*75*sinTheta
Work done by pulling=140*75
Kinetic energy=1/2 m v^2
so
140*75=1/2 mv^2+mg*75*sinTheta
and you solve for v.
To find the speed of the wagon after moving 75 m up the hill, we can use the principle of work and energy.
First, let's identify the given information:
- Mass of the wagon (m): 40.0 kg
- Inclination angle (θ): 17.9°
- Tension in the rope (T): 140 N
- Distance traveled up the hill (d): 75 m
Now, let's start by calculating the work done on the wagon. The work done on an object is given by the formula:
Work = Force × Distance × cos(θ)
In this case, the force applied is the tension in the rope (T), the distance is the distance traveled up the hill (d), and the angle is the inclination angle (θ). Substituting these values into the formula, we get:
Work = 140 N × 75 m × cos(17.9°)
Next, we can calculate the gravitational potential energy gained by the wagon as it moves up the hill. The change in gravitational potential energy can be calculated using the formula:
ΔPE = m × g × Δh
Since the wagon starts from rest at the bottom of the hill, the initial gravitational potential energy is zero. The final gravitational potential energy gained by the wagon is given by the mass of the wagon (m), acceleration due to gravity (g), and the change in height (Δh).
In this case, the change in height is given by:
Δh = d × sin(θ)
Substituting the values into the formula, we have:
ΔPE = 40.0 kg × 9.8 m/s^2 × (75 m × sin(17.9°))
According to the principle of work and energy, the work done on an object is equal to the change in its energy. Therefore, we can equate the work done on the wagon to the change in its gravitational potential energy:
Work = ΔPE
140 N × 75 m × cos(17.9°) = 40.0 kg × 9.8 m/s^2 × (75 m × sin(17.9°))
Now, we can solve this equation to calculate the speed of the wagon. Rearranging the equation to isolate the velocity (v), we get:
v = √((2 × ΔPE) / m)
Substituting the values into the formula, we can find the velocity:
v = √((2 × (40.0 kg × 9.8 m/s^2 × (75 m × sin(17.9°)))) / 40.0 kg)
By evaluating this expression, we can find the speed of the wagon after moving 75 m up the hill.