A golf bounces down a flight of steel stairs, striking several steps on the way down, but never hitting the edge of a step. The ball starts at the top step with a vertical velocity component of zero. If all the collisions with the stairs are elastic, and if the vertical height of the staircase is 3.00 m, determine the bounce height when the ballreaches the bottom of the stairs. Neglect air resistance.
The change of gravitational potental energy, mgh, goes into KE due to vertical motion. With no losses on bouncing (there will be in reality), the top of each bounce will be at the original height.
Well, I guess the golf ball really wanted to go on a stair-focused adventure, huh?
Anyway, let's break it down. The ball starts at the top with a vertical velocity component of zero, meaning it's not going up or down initially. As it bounces down the stairs, it'll gain speed due to gravity, but we need to figure out how high it bounces back up at the end.
Since all collisions with the stairs are elastic, we know energy is conserved. This means that the total energy of the ball remains constant throughout its journey. So, let's use the conservation of energy to solve this.
At the top step, the ball only has potential energy, given by mgh, where m is the mass of the ball, g is the gravitational acceleration, and h is the height of the staircase.
As the ball reaches the bottom step, it will only have kinetic energy, since its potential energy is now zero. Let's call this kinetic energy K.
By the conservation of energy, we can equate the initial potential energy to the final kinetic energy:
mgh = K
We also know that the total mechanical energy (potential energy + kinetic energy) remains constant, as it is conserved. So we can write:
mgh = mgh' + K'
Here, h' represents the bounce height, and K' represents the final kinetic energy after the bounce. Since the ball doesn't hit the edge of a step, we can assume the bounce height is half the staircase height.
Therefore, we have:
mgh = mgh' + K'
Since it's an elastic collision, we can also say that the final kinetic energy K' is equal to the initial kinetic energy before the bounce, which is zero:
mgh = mgh' + 0
So, mgh = mgh'
We can now cancel the mass and g from both sides of the equation:
h = h'
Voila! The bounce height is equal to the original height of the staircase, which is 3.00 m in this case. So, the golf ball will bounce back up to a height of 3.00 m. Quite the bouncy adventure, eh?
To determine the bounce height of the golf ball when it reaches the bottom of the stairs, we can use the concept of conservation of energy.
Given:
Vertical height of the staircase = 3.00 m
1. First, let's determine the initial velocity of the golf ball when it is dropped from the top step. Since the ball has a vertical velocity component of zero, we can assume that the initial velocity is also zero.
2. As the ball drops down the staircase, it loses gravitational potential energy (PE) and gains an equal amount of kinetic energy (KE). The total mechanical energy (KE + PE) remains constant throughout the motion.
3. At each collision with a step, the ball loses some kinetic energy due to an elastic collision. However, the loss of energy is purely horizontal since the ball never hits the edge of a step. The vertical component of velocity (V) remains the same.
4. The maximum height (h) the ball reaches after each bounce can be calculated using the conservation of energy equation: PE = KE.
PE = mgh (where m is the mass of the ball, g is the gravitational acceleration, and h is the height)
KE = 1/2mv² (where v is the velocity)
Therefore, mgh = 1/2mv²
Canceling out the mass (m) from both sides of the equation, we get:
gh = 1/2v²
Since the vertical component of velocity (V) remains the same throughout the motion, we can rewrite the equation as:
g * 3m = 1/2V²
5. Rearranging the equation, we get:
V = √(2g * 3)
6. Now, we can calculate the bounce height when the ball reaches the bottom of the stairs using the same equation as before:
gh = 1/2V²
Substituting the values of g and V into the equation, we get:
(9.8 m/s²) * h = 1/2(√(2 * 9.8 m/s² * 3))^2
Simplifying the equation, we get:
9.8h = 1/2(√(58.8))^2
9.8h = 1/2(7.67)^2
9.8h = 1/2(58.89)
9.8h = 29.44
h ≈ 3.00 m
Therefore, the bounce height of the golf ball when it reaches the bottom of the stairs is approximately 3.00 meters, the same as the vertical height of the staircase.
To find the bounce height of the golf ball when it reaches the bottom of the stairs, we can use the concept of conservation of mechanical energy.
Let's assume that the first bounce of the golf ball takes place at a certain height h above the bottom of the stairs.
To calculate the initial kinetic energy of the first bounce, we need to know the potential energy at the top of the staircase. The potential energy can be calculated using the formula:
Potential energy = mgh
Where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the staircase (3.00 m in this case).
Since the ball starts with a vertical velocity component of zero, the initial kinetic energy is also zero.
Therefore, the initial mechanical energy (E₁) can be expressed as:
E₁ = Potential energy + Kinetic energy
= mgh + 0
= mgh
According to the principle of conservation of mechanical energy, the total mechanical energy (E₂) at the bottom of the stairs (after the bounce) will be equal to the initial mechanical energy (E₁). However, the potential energy will be converted to kinetic energy, and vice versa.
For an elastic collision, the total mechanical energy is conserved, and therefore:
E₂ = Potential energy + Kinetic energy
= mgh + 0
= mgh
Since the ball never hits the edge of a step, the kinetic energy after each bounce remains zero.
Now, since the golf ball bounces up to a certain height h above the bottom of the stairs, the potential energy at that height can be calculated using the formula:
Potential energy = mgh
Thus, the final mechanical energy (E₂) at that height will be the sum of the potential energy (mgh) and the kinetic energy (0), which gives:
E₂ = mgh
Since E₁ = E₂, we can equate the expressions for E₁ and E₂ to find:
mgh = mgh
Simplifying the equation, we find that the mass (m) and gravity (g) cancel out, leaving:
h = h
This means that the bounce height of the golf ball when it reaches the bottom of the stairs will be equal to the height of the staircase, which is 3.00 m in this case.
Therefore, the bounce height will be 3.00 meters.