A plastic cup,15.0 cm in height,is filled to the rim with water.A small hole is punctured in the cup at a height 9.00 cm above the bottom of the cup and is allowing water to spew out.At what distance does the water land from the bottom of the cup?

Question

Quizplus · Accepted Answer

We can use the fact that the water will follow a parabolic path as it exits the hole. We can use the equations of motion to find the distance the water lands from the bottom of the cup. 
First, we can find the initial velocity of the water as it exits the hole. We can use the fact that the water is at the same level as the hole when it exits, so the initial vertical velocity is 0. We can use conservation of energy to find the initial velocity: 
mgh = 1/2 mv^2 
where m is the mass of the water, g is the acceleration due to gravity, h is the height of the hole above the bottom of the cup, and v is the initial velocity. We know the mass of the water is equal to the volume of water (since the density of water is 1 g/cm^3), so 
m = V = pi r^2 h 
where r is the radius of the cup and h is the height of the water in the cup. 
Plugging in the given values, we get 
v = sqrt(2gh) = sqrt(2*9.81 m/s^2 * (15.0 cm - 9.00 cm)) = 2.67 m/s 
Now we can use the fact that the horizontal velocity is constant to find the distance the water lands from the bottom of the cup. 
We can use the equation 
d = v_horizontal * t 
where d is the distance the water lands from the bottom of the cup, v_horizontal is the horizontal velocity of the water (which is the same as the initial horizontal velocity of the water), and t is the time it takes for the water to hit the ground. 
We can use the fact that the time it takes for an object to fall a certain distance is given by 
h = 1/2 g t^2 
where g is the acceleration due to gravity and h is the distance the object falls. We want to find the time it takes for the water to fall a distance of (15.0 - 9.00) cm = 6.00 cm, so we get 
t = sqrt(2h/g) = sqrt(2*0.06 m / 9.81 m/s^2) = 0.11 s 
Plugging in the given values, we get 
d = v_horizontal * t = 2.67 m/s * 0.11 s = 0.2937 m = 14.6 cm 
So the water lands 14.6 cm from the bottom of the cup. 
Therefore, the answer is C.

A Plastic Cup,15