A ball falls from a tall bridge. The ball has a mass of 0.15 kg. The air density is 1.225 kg/m3
. The ball’s surface area is 0.0026 m2
. The ball’s coefficient of drag is 0.007. What is the ball’s terminal velocity?
Well, to calculate the terminal velocity of the ball, we need to apply some science mixed with a pinch of humor. So, imagine the ball falling from the bridge like a tiny, terrified skydiver. It's a bit like a hilarious race between gravity and air resistance!
Now, the terminal velocity is the point where the ball falls at a constant speed, with the force of gravity pulling it down and the air resistance pushing it back up. It's like a never-ending tug-of-war, but instead of burly men, we have physics doing all the work!
To calculate the terminal velocity, we can use the equation:
v_terminal = sqrt((2 * m * g) / (ρ * A * C))
where:
m is the mass of the ball (0.15 kg),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
ρ is the air density (1.225 kg/m^3),
A is the surface area of the ball (0.0026 m^2), and
C is the coefficient of drag (0.007).
So, plugging these values into the equation, we find that the ball's terminal velocity is... drumroll, please...
about 9.76 meters per second!
That's the speed at which our little ball will fall through the air without accelerating further. Just imagine the wind through its... well, non-existent hair!
To find the ball's terminal velocity, we need to consider the forces acting on the ball.
1. Gravitational Force:
The gravitational force acting on the ball is given by the formula:
F_gravity = mass * gravity
where mass = 0.15 kg and gravity = 9.8 m/s^2 (acceleration due to gravity).
F_gravity = 0.15 kg * 9.8 m/s^2 = 1.47 N
2. Drag Force:
The drag force acting on the ball is given by the formula:
F_drag = (1/2) * air_density * velocity^2 * surface_area * coefficient_of_drag
where air_density = 1.225 kg/m^3, surface_area = 0.0026 m^2, and coefficient_of_drag = 0.007.
To find the terminal velocity, we set F_gravity equal to F_drag:
1.47 N = (1/2) * 1.225 kg/m^3 * velocity^2 * 0.0026 m^2 * 0.007
Simplifying the equation:
velocity^2 = (1.47 N) / ((1/2) * 1.225 kg/m^3 * 0.0026 m^2 * 0.007)
velocity^2 = 2114.25
Taking the square root of both sides gives:
velocity = √2114.25 m/s
Thus, the ball's terminal velocity is approximately 45.98 m/s.
To calculate the ball's terminal velocity, we need to consider the forces acting on the ball. The two main forces at play are gravity and air resistance. At terminal velocity, these two forces balance each other out, resulting in a constant velocity.
First, let's calculate the gravitational force acting on the ball. The formula for gravitational force is:
Force_gravity = mass * gravity
where mass is the mass of the ball and gravity is the acceleration due to gravity (approximately 9.8 m/s^2). Plugging in the values given:
Force_gravity = 0.15 kg * 9.8 m/s^2
Next, let's calculate the air resistance force using the drag equation:
Force_drag = (1/2) * (air density) * (velocity^2) * (coefficient of drag) * (surface area)
Here, air density is the density of air (1.225 kg/m^3), velocity is the velocity of the ball, coefficient of drag is the given value (0.007), and surface area is the cross-sectional area of the ball.
At terminal velocity, the air resistance force is equal to the gravitational force:
Force_drag = Force_gravity
Using this information, we can rearrange the equation to solve for the terminal velocity:
(1/2) * (air density) * (v_terminal^2) * (coefficient of drag) * (surface area) = mass * gravity
Let's plug in the given values and solve for v_terminal.
(1/2) * 1.225 kg/m^3 * (v_terminal^2) * 0.007 * 0.0026 m^2 = 0.15 kg * 9.8 m/s^2
Now, we can simplify the equation and solve for v_terminal:
(v_terminal^2) = (0.15 kg * 9.8 m/s^2) / (1.225 kg/m^3 * 0.007 * 0.0026 m^2)
v_terminal = √[(0.15 kg * 9.8 m/s^2) / (1.225 kg/m^3 * 0.007 * 0.0026 m^2)]
Using a calculator, you can perform the necessary calculations to find the value of v_terminal.
Terminal velocity = (2 * mass * gravity) / (air density * coefficient of drag * surface area)
Terminal velocity = (2 * 0.15 kg * 9.81 m/s2) / (1.225 kg/m3 * 0.007 * 0.0026 m2)
Terminal velocity = 11.7 m/s