An 8 kg bowling ball creates air resistance according to F(air)=2.14v. The bowling ball is rolled off the top of the sears tower.
a) Calculate the acceleration (magnitude and direction at
1) The top of the building____
2)when the ball reaches 5.6 m/s_____
b) What is the terminal velocity____?
To calculate the acceleration of the bowling ball, we need to consider both the force due to gravity and the force due to air resistance.
a)
1) At the top of the building, the ball is not yet in motion. Therefore, there is no acceleration. This means the magnitude of acceleration is 0 m/s², and the direction is not applicable.
2) When the ball reaches a velocity of 5.6 m/s, we can use the following equation to calculate the net force acting on the ball:
Net Force = Force of Gravity - Force of Air Resistance
The force of gravity can be calculated using Newton's second law of motion:
Force of Gravity = mass × acceleration due to gravity
Given that the mass of the bowling ball is 8 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the force of gravity:
Force of Gravity = 8 kg × 9.8 m/s² = 78.4 N
The force of air resistance is given by the equation F(air) = 2.14v, where v is the velocity of the ball. At 5.6 m/s, we can calculate the force of air resistance:
Force of Air Resistance = 2.14 × 5.6 = 11.9 N
Now, we can substitute these values into the equation for net force:
Net Force = 78.4 N - 11.9 N = 66.5 N
Since we know that force = mass × acceleration, we can rearrange the equation to solve for the acceleration:
Acceleration = Net Force / mass = 66.5 N / 8 kg = 8.31 m/s²
Therefore, the magnitude of acceleration when the ball reaches 5.6 m/s is 8.31 m/s². The direction of the acceleration is downward, as the net force is less than the force of gravity.
b) The terminal velocity is the maximum velocity an object can reach when the force of air resistance equals the force of gravity. In other words, it is the point at which the net force on the object becomes zero. At terminal velocity, there is no acceleration.
For the bowling ball, the force of air resistance is given by F(air) = 2.14v, and the force of gravity is given by mass × acceleration due to gravity.
Setting the forces equal to each other:
2.14v = mass × acceleration due to gravity
Substituting the known values:
2.14v = 8 kg × 9.8 m/s²
Simplifying the equation:
v = (8 kg × 9.8 m/s²) / 2.14
v ≈ 36.26 m/s
Therefore, the terminal velocity of the bowling ball is approximately 36.26 m/s.