A boy on board a cruise ship drops a 30g marble into the ocean. If the resistive force proportionality constant is 0.5kg/s, what is the terminal speed of the marble in m/s?
1st: convert 30gto kg=.03kg
2nd: .03kg*9.81m/s^2=.294kg*m/s^2
3rd: answer .(94kg*m/s^2)÷(.5kg/s)=.589m/s
To find the terminal speed of the marble, we need to consider the balance between the gravitational force pulling it downwards and the resistive force acting against it.
The gravitational force pulling the marble downwards can be calculated using the equation:
F_gravity = m * g
Where:
m = mass of the marble (30g = 0.03kg)
g = acceleration due to gravity (9.8 m/s²)
F_gravity = 0.03kg * 9.8 m/s² = 0.294 N
The resistive force acting against the marble can be calculated using the equation:
F_resistive = k * v
Where:
k = resistive force proportionality constant (0.5 kg/s)
v = velocity of the marble
Now, at terminal velocity, the gravitational force equals the resistive force, so we can set these two equations equal to each other:
F_gravity = F_resistive
0.294 N = (0.5 kg/s) * v
To solve for v, we rearrange the equation:
v = 0.294 N / (0.5 kg/s)
v = 0.588 m/s
Therefore, the terminal speed of the marble is 0.588 m/s.
To find the terminal speed of the marble, we first need to understand the forces acting on the marble as it falls through the ocean.
When an object falls through a fluid, such as water, two main forces are acting on it:
1. Gravitational force (Weight): This force pulls the marble downward and is equal to its mass (m) multiplied by the acceleration due to gravity (g).
Weight (W) = m * g
2. Resistive force (Drag): This force opposes the motion of the marble and increases as the speed of the marble increases. It can be calculated using the formula:
Drag force (D) = k * v
where k is the proportionality constant and v is the velocity of the marble.
At terminal speed, the drag force becomes equal in magnitude to the weight, resulting in no further acceleration and a constant velocity. Therefore, at terminal speed, the weight of the marble is equal to the drag force:
m * g = k * v
To find the terminal speed v, we rearrange the equation:
v = (m * g) / k
Given that the mass of the marble is 30g (or 0.03kg) and the proportionality constant k is 0.5kg/s, we can substitute these values into the equation to find the terminal speed v:
v = (0.03kg * 9.8m/s^2) / 0.5kg/s
v = 0.588 m/s
So, the terminal speed of the marble in the ocean is approximately 0.588 m/s.