In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is 1.12 m. The mug slides off the counter and strikes the floor 2.00 m from the base of the counter.
So, what's the question?
How far did the shattered glass spread?
How big was the beer puddle?
What was the bartender's name?
To determine the time it takes for the mug to strike the floor, we can use the principles of projectile motion.
First, let's consider the vertical motion of the mug. The equation for the vertical displacement is given by:
Δy = v₀y * t + (1/2) * g * t²
Where:
- Δy is the vertical displacement (1.12 m, the height of the counter)
- v₀y is the initial vertical velocity (which we need to find)
- t is the time it takes for the mug to hit the floor
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Since the mug starts with zero initial vertical velocity, the equation simplifies to:
Δy = (1/2) * g * t²
Substituting the values:
1.12 = (1/2) * 9.8 * t²
Next, let's consider the horizontal motion of the mug. The equation for the horizontal displacement is given by:
Δx = v₀x * t
Where:
- Δx is the horizontal displacement (2.00 m, distance from the base of the counter)
- v₀x is the initial horizontal velocity (which we need to find)
- t is the same time from the vertical motion
Without any horizontal forces acting on the mug, the initial horizontal velocity v₀x remains constant. Therefore:
Δx = v₀x * t
Substituting the values:
2 = v₀x * t
Now, we can solve these two equations simultaneously to find both t and v₀x.
From the equation Δx = v₀x * t, we can rearrange to solve for v₀x:
v₀x = Δx / t
Plugging in the values:
v₀x = 2 / t
Now, substitute this value for v₀x into the equation Δy = (1/2) * g * t²:
1.12 = (1/2) * 9.8 * t²
Rearranging the equation:
t² = (2 * 1.12) / 9.8
Solving for t:
t = √(2 * 1.12 / 9.8)
Calculating the value of t gives us the time it takes for the mug to hit the floor.