A 38.2 kg wagon is towed up a hill inclined at 19° with respect to the horizontal. The tow rope is parallel to the incline and exerts a force of 140 N on the wagon. Assume that the wagon starts from rest at the bottom of the hill, and disregard friction.How fast is the wagon going after moving 30.0 m up the hill?
12.77
To find the speed of the wagon after it has moved up the hill, we can use the principle of work-energy.
1. First, let's calculate the work done by the force exerted by the tow rope on the wagon. The formula for work is given by:
Work = Force x Distance x cos(theta)
where:
- Force is the force exerted by the tow rope (140 N),
- Distance is the distance traveled up the hill (30.0 m), and
- theta is the angle between the force and the direction of motion (19°).
So, plugging in the values:
Work = 140 N x 30.0 m x cos(19°)
2. Next, let's calculate the change in potential energy of the wagon as it moves up the hill. The formula for potential energy is given by:
Potential Energy = Mass x Gravitational Acceleration x Height
where:
- Mass is the mass of the wagon (38.2 kg),
- Gravitational Acceleration is the acceleration due to gravity (9.8 m/s²), and
- Height is the vertical distance traveled up the hill (given by the inclined plane).
The height can be calculated as:
Height = Distance x sin(theta)
Plugging in the values:
Height = 30.0 m x sin(19°)
Potential Energy = 38.2 kg x 9.8 m/s² x (30.0 m x sin(19°))
3. According to the principle of work-energy, the work done on an object is equal to the change in its mechanical energy. In this case, the mechanical energy is the sum of kinetic energy and potential energy. Since the wagon starts from rest, its initial kinetic energy is zero. Therefore:
Work = Change in Potential Energy + Final Kinetic Energy
Rearranging the equation, we get:
Final Kinetic Energy = Work - Change in Potential Energy
4. Kinetic energy is given by the formula:
Kinetic Energy = 0.5 x Mass x Velocity²
Plugging in the values, we can solve for velocity:
0.5 x Mass x Velocity² = Work - Change in Potential Energy
Velocity = √(2 x (Work - Change in Potential Energy) / Mass)
5. Finally, plug in the calculated values into the equation and solve for velocity:
Velocity = √(2 x (Work - Change in Potential Energy) / Mass)