Answer :
Assuming a drag coefficient of 0.5, the table tennis ball will rise to the surface of the swimming pool with a velocity of approximately 9.24 m/s.
The velocity at which the table tennis ball rises to the surface of the swimming pool can be determined using the concept of terminal velocity. Terminal velocity is the constant velocity that an object reaches when the drag force acting on it equals the gravitational force pulling it downwards.
To calculate the velocity, we need to consider the weight of the ball, the drag coefficient (cd), and the diameter of the ball.
Given:
- Diameter of the ball = 38.1 mm
- Weight of the ball = 0.0245 N
Step 1: Convert the diameter to meters:
The diameter of the ball is given in millimeters (mm), so we need to convert it to meters. Since there are 1000 millimeters in a meter, the diameter in meters is:
38.1 mm = 38.1 / 1000 = 0.0381 m
Step 2: Calculate the cross-sectional area of the ball:
The cross-sectional area of a sphere can be calculated using the formula: A = πr^2, where r is the radius of the sphere. The radius is half the diameter, so:
radius = 0.0381 m / 2 = 0.01905 m
cross-sectional area = π(0.01905 m)^2
Step 3: Calculate the drag force:
The drag force is given by the formula: drag force = 0.5 * cd * ρ * A * velocity^2, where cd is the drag coefficient and ρ is the density of the fluid (water).
Since the ball is in water, the density of water is 1000 kg/m^3.
Step 4: Set up the equation:
At terminal velocity, the drag force equals the weight of the ball. So we can set up the equation:
0.5 * cd * ρ * A * velocity^2 = weight
Step 5: Solve for velocity:
Substitute the given values into the equation and solve for velocity:
0.5 * cd * 1000 kg/m^3 * cross-sectional area * velocity^2 = 0.0245 N
Simplify the equation:
0.5 * cd * 1000 kg/m^3 * π(0.01905 m)^2 * velocity^2 = 0.0245 N
Solve for velocity:
velocity^2 = 0.0245 N / (0.5 * cd * 1000 kg/m^3 * π(0.01905 m)^2)
velocity^2 = 0.0245 N / (0.5 * cd * 1000 kg/m^3 * 0.00114461 m^2)
velocity^2 = 42.71 N/m^2 / cd
Taking the square root of both sides gives:
velocity = √(42.71 N/m^2 / cd)
Step 6: Substitute the value of cd:
Unfortunately, the value of the drag coefficient (cd) is not provided in the question. The drag coefficient depends on the shape and surface characteristics of the ball. It would be helpful to have this information in order to calculate the velocity accurately.
However, assuming a typical drag coefficient for a sphere in water is around 0.5, we can substitute this value into the equation and solve for the velocity:
velocity = √(42.71 N/m^2 / 0.5)
velocity = √(85.42 N/m^2)
velocity ≈ √(85.42) m/s
velocity ≈ 9.24 m/s
Therefore, assuming a drag coefficient of 0.5, the table tennis ball will rise to the surface of the swimming pool with a velocity of approximately 9.24 m/s.
Learn more about drag coefficient
https://brainly.com/question/23942493
#SPJ11
