Pablo is playing with a hula-hoop (diameter of 112 cm and mass of 858 g). While playing with the hula-hoop, Pablo accidently lets it roll down the driveway towards the street. The driveway is 15.0 m long and has a constant slope of 6 degrees. The hula-hoop starts with a linear speed of 1.30 m/s. Find the linear final velocity of the hula hoop and how long it takes for the hula hoop to reach the bottom of the driveway.
To find the linear final velocity and the time it takes for the hula hoop to reach the bottom of the driveway, we can use the principles of physics, specifically kinematics and energy conservation.
First, let's calculate the angle of the slope in radians. We know that θ (theta) in degrees is given as 6 degrees. To convert degrees to radians, we use the following formula:
Angle in radians = Angle in degrees × π / 180
Plugging in the values, we have:
θ (radians) = 6 × π / 180 = 0.10472 radians (rounded to 5 decimal places)
Next, we need to find the height difference between the starting point and the bottom of the driveway. The hula hoop rolls down the driveway, so the vertical distance traveled is equal to the height difference between the start and end points. We can use trigonometry to find this distance.
Height = Length of the driveway × sin(θ)
Height = 15.0 m × sin(0.10472)
Height ≈ 1.561 m (rounded to 3 decimal places)
Now, let's find the final linear velocity of the hula hoop using energy conservation. Since there is no significant friction involved, the total mechanical energy of the hula hoop is conserved.
Initial mechanical energy = Final mechanical energy
The initial mechanical energy is given by:
Initial mechanical energy = (1/2) × mass × initial linear velocity^2
Plugging in the values, we have:
Initial mechanical energy = (1/2) × 0.858 kg × (1.30 m/s)^2
Initial mechanical energy ≈ 1.581 J (rounded to 3 decimal places)
The final mechanical energy only consists of potential energy since the hula hoop has reached the bottom of the driveway:
Final mechanical energy = mass × gravitational acceleration × height
The gravitational acceleration, g, is approximately 9.8 m/s^2. Plugging in the values, we have:
Final mechanical energy = 0.858 kg × 9.8 m/s^2 × 1.561 m
Final mechanical energy ≈ 13.347 J (rounded to 3 decimal places)
Now, equating the initial and final mechanical energies, we have:
(1/2) × mass × initial linear velocity^2 = mass × gravitational acceleration × height
Simplifying the equation, we get:
(1/2) × initial linear velocity^2 = gravitational acceleration × height
Next, solve for the initial linear velocity^2:
initial linear velocity^2 = 2 × gravitational acceleration × height
Plugging in the values, we have:
initial linear velocity^2 = 2 × 9.8 m/s^2 × 1.561 m
initial linear velocity^2 ≈ 30.715 m^2/s^2 (rounded to 3 decimal places)
To find the linear final velocity, take the square root of the initial linear velocity^2:
linear final velocity = √(initial linear velocity^2)
linear final velocity ≈ √30.715 m^2/s^2 ≈ 5.527 m/s (rounded to 3 decimal places)
Finally, to find the time it takes for the hula hoop to reach the bottom of the driveway, we can use kinematic equations.
The equation we'll use is:
Final linear velocity = Initial linear velocity + acceleration × time
Since the acceleration in this case is due to gravity and is approximately 9.8 m/s^2 downward, the equation becomes:
5.527 m/s = 1.30 m/s + 9.8 m/s^2 × time
Simplifying the equation, we have:
4.227 m/s = 9.8 m/s^2 × time
Now, solve for time:
time = 4.227 m/s / 9.8 m/s^2
time ≈ 0.432 s (rounded to 3 decimal places)
Therefore, the linear final velocity of the hula hoop is approximately 5.527 m/s, and it takes about 0.432 seconds for the hula hoop to reach the bottom of the driveway.