A baseball of mass 0.25 kg is hit so hard that it bounces on the ground and winds up in a spectator’s seat. If the ball leaves the ground at a speed 8.2 m/s and lands in the seat with a speed of 2.3 m/s, at what height, in metres relative to the ground, is the spectator?
The height of the spectator relative to the ground can be calculated using the equation:
h = (vf^2 - vi^2) / (2 * g)
where h is the height, vf is the final velocity, vi is the initial velocity, and g is the acceleration due to gravity (9.8 m/s^2).
Plugging in the given values, we get:
h = (2.3^2 - 8.2^2) / (2 * 9.8)
h = -7.2 m
Since the height is negative, this means that the spectator is 7.2 m below the ground.
Well, let's solve this bouncing baseball mystery! We can use the principle of conservation of energy. The initial kinetic energy of the ball when it leaves the ground is given by (1/2)mv^2, where m is the mass and v is the initial velocity. The final kinetic energy of the ball when it lands in the seat is given by (1/2)mv^2, where v is the final velocity.
Now, we know that the initial kinetic energy is converted into potential energy when the baseball reaches its maximum height. So, we can equate the initial kinetic energy to the potential energy at maximum height.
(1/2)mv^2 = mgh
Here, m is the mass of the baseball, v is the initial velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height.
We can rearrange the equation to solve for h:
h = (1/2) v^2 /(g)
Plugging in the given values, we have:
h = (1/2) x (8.2 m/s)^2 / (9.8 m/s^2)
Calculating this, we find:
h ≈ 3.43 meters
So, the spectator is approximately 3.43 meters above the ground. I hope that answers your question and didn't bounce over your head with all the physics!
To solve this problem, we can use the principles of conservation of mechanical energy.
Step 1: Find the initial potential energy of the baseball.
The initial potential energy (PEi) of the baseball can be found using the formula:
PEi = m * g * hi
where m is the mass of the baseball (0.25 kg), g is the acceleration due to gravity (9.8 m/s^2), and hi is the initial height.
Step 2: Find the final potential energy of the baseball.
The final potential energy (PEf) of the baseball can be found using the formula:
PEf = m * g * hf
where hf is the final height.
Step 3: Find the work done by gravitational force.
The work done by the gravitational force on the baseball is given by:
W = PEf - PEi
Step 4: Find the kinetic energy of the baseball.
The kinetic energy (KE) of the baseball is given by:
KE = (1/2) * m * v^2
where v is the final velocity (2.3 m/s).
Step 5: Find the work done against the gravitational force.
The work done against the gravitational force is the difference between the kinetic energy and the work done by the gravitational force:
Work against gravity = KE - W
Step 6: Find the height relative to the ground.
The height relative to the ground (hf) can be found using the formula:
hf = Work against gravity / (m * g)
Substituting the given values:
m = 0.25 kg
g = 9.8 m/s^2
v = 2.3 m/s
we can calculate hf as follows:
PEi = m * g * hi = 0.25 kg * 9.8 m/s^2 * hi
PEf = m * g * hf = 0.25 kg * 9.8 m/s^2 * hf
W = PEf - PEi
W = (0.25 kg * 9.8 m/s^2 * hf) - (0.25 kg * 9.8 m/s^2 * hi)
W = 0.25 kg * 9.8 m/s^2 * (hf - hi)
KE = (1/2) * m * v^2
KE = (1/2) * 0.25 kg * (2.3 m/s)^2
Work against gravity = KE - W
Work against gravity = (1/2) * 0.25 kg * (2.3 m/s)^2 - 0.25 kg * 9.8 m/s^2 * (hf - hi)
hf = Work against gravity / (m * g)
hf = [(1/2) * 0.25 kg * (2.3 m/s)^2 - 0.25 kg * 9.8 m/s^2 * (hf - hi)] / (0.25 kg * 9.8 m/s^2)
Now, we can solve this equation to find the height relative to the ground (hf).
To find the height of the spectator, we need to calculate the change in potential energy of the baseball as it bounces from the ground to the seat.
First, let's calculate the initial kinetic energy of the baseball as it leaves the ground. We can use the formula:
Kinetic energy = (1/2) * mass * velocity^2
Given:
Mass of the baseball (m) = 0.25 kg
Initial velocity of the baseball (v_initial) = 8.2 m/s
Plugging in the values, we get:
Initial kinetic energy = 0.5 * 0.25 kg * (8.2 m/s)^2
Next, let's calculate the final kinetic energy of the baseball as it lands in the seat. Using the same formula:
Final velocity of the baseball (v_final) = 2.3 m/s
Plugging in the values, we get:
Final kinetic energy = 0.5 * 0.25 kg * (2.3 m/s)^2
Now, let's calculate the change in kinetic energy of the baseball:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Next, we will use the conservation of energy principle, which states that the change in potential energy of an object is equal to the change in kinetic energy. Therefore, we can equate the change in potential energy to the change in kinetic energy calculated above.
Change in potential energy = Change in kinetic energy
Since potential energy is given by the formula:
Potential energy = mass * gravity * height
where gravity is approximately 9.8 m/s^2, and the mass of the baseball is 0.25 kg, we can rewrite the equation as:
0.25 kg * 9.8 m/s^2 * height = Change in kinetic energy
Now, we can solve for the height (h):
Height (h) = (Change in kinetic energy) / (0.25 kg * 9.8 m/s^2)
By substituting the value of the change in kinetic energy calculated earlier and solving for h, we can determine the height of the spectator above the ground.