K-1 whitewater kayak has a volume of 225 liters (0.225 m3
) and a mass of 10 kg. Sean
wonders if the kayak is big enough for him to use. Sean’s mass is 70 kg. What percent of
the kayak will be submerged when Sean sits in it? Assume that the water density is 1.00
g/cm3
(1000 kg/m3
).

Question

K-1 whitewater kayak has a volume of 225 liters (0.225 m3
) and a mass of 10 kg. Sean 
wonders if the kayak is big enough for him to use. Sean’s mass is 70 kg. What percent of 
the kayak will be submerged when Sean sits in it? Assume that the water density is 1.00 
g/cm3
(1000 kg/m3
).

Vocaloid · Answer

Using Archimedes' principle:

The buoyant force exerted by the water = the weight of the displaced water.

$$F_{b}=m_{w}g=(ρ_{w})V_{w}(g)$$

Where the subscript w refers to the amount of water displaced by the boat. The amount of water displaced by the boat is also equal to the volume of the boat submerged, since the boat displaces water when it's submerged. Let's change Vw to Vs to stand for volume of the boat submerged.

$$F_{b}=m_{w}g=(ρ_{w})V_{s}(g)$$

For the kayak to float, the gravitational force on the boat must be equal to the buoyant force.

$$F_{b}=(ρ_{w})V_{s}(g)=m_{b}(g)$$

g cancels out from both sides:

$$F_{b}=(ρ_{w})V_{s}=m_{b}$$

so you can plug in the mass of the boat (make sure to also add in Sean's weight), and the density of water, and then solve for Vs, the volume submerged. Since it's asking for what percent of the total kayak's volume is submerged, divide Vs by the total kayak volume and convert to a percent.