In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is 1.18 m. The mug slides off the counter and strikes the floor 1.20 m from the base of the counter.
(a) With what velocity did the mug leave the counter?
(b) What was the direction of the mug's velocity just before it hit the floor?
To calculate the velocity of the mug as it leaves the counter and the direction of its velocity just before it hits the floor, we can use the principles of kinematics.
(a) To find the velocity of the mug as it leaves the counter, we can use the equation of motion:
v² = u² + 2as
Where:
- v is the final velocity (which we want to find)
- u is the initial velocity (which is 0 since the mug initially starts from rest)
- a is the acceleration (which we can assume to be gravity, approximately 9.8 m/s²)
- s is the distance traveled (which is the height of the counter, 1.18 m)
Plugging in the values into the equation:
v² = 0² + 2 * 9.8 * 1.18
v² = 22.032
v ≈ √22.032
v ≈ 4.7 m/s
Therefore, the mug left the counter with a velocity of approximately 4.7 m/s.
(b) To determine the direction of the mug's velocity just before it hits the floor, we need to consider the mug's motion after leaving the counter. Since the mug is affected only by the force of gravity, it will follow a parabolic path known as a projectile motion.
At the moment the mug hits the floor, the velocity vector will be pointing downwards since gravity is pulling the mug towards the ground. Thus, the direction of the mug's velocity just before it hits the floor is downward, perpendicular to the floor.
Note: It's important to mention that the mug's motion in the horizontal direction after leaving the counter is not considered in this particular problem, as we are only interested in determining its velocity and direction just before hitting the floor vertically.
(a) how long does it take to fall 1.18m?
4.9t^2 = 1.18
now you know that vt = 1.20
(b) you have the horizontal speed vx
The vertical speed is vy = -9.8t
so the velocity is vx i + vy j
and tanθ = vy / vx