A student drops the same 10 kilogram bowling ball three separate times. She drops the ball from heights of 1 meter, 2 meters, and 3 meters. Which graph correctly shows the ball’s kinetic energy when dropped from each height?(1 point)
Responses
Unfortunately, I cannot provide graphs as a text-based AI. However, I can describe the correct graph for you.
The kinetic energy of an object is given by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.
Since the mass of the bowling ball is 10 kilograms and the height from which it is dropped is different each time, we need to calculate the velocity (v) for each height.
When the ball is dropped from a height of 1 meter, it will have a certain velocity when it hits the ground. We can calculate this using the formula v = sqrt(2 * g * h), where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height (1 meter). Let's call this velocity v1.
Similarly, when the ball is dropped from a height of 2 meters, it will have another velocity when it hits the ground. We can calculate this using the same formula, but with h = 2 meters. Let's call this velocity v2.
Finally, when the ball is dropped from a height of 3 meters, it will have another velocity when it hits the ground. We can calculate this using the same formula, but with h = 3 meters. Let's call this velocity v3.
Now, let's consider the kinetic energy of the ball when dropped from each height.
When the ball is dropped from a height of 1 meter, its kinetic energy will be KE1 = 1/2 * 10 * v1^2.
When the ball is dropped from a height of 2 meters, its kinetic energy will be KE2 = 1/2 * 10 * v2^2.
When the ball is dropped from a height of 3 meters, its kinetic energy will be KE3 = 1/2 * 10 * v3^2.
To graph these kinetic energies correctly, you would plot the heights (1 meter, 2 meters, 3 meters) on the x-axis and the corresponding kinetic energies (KE1, KE2, KE3) on the y-axis. The graph should show a linear relationship, with the kinetic energy increasing as the height increases.
Note that the exact shape of the graph would depend on the specific values of v1, v2, and v3, but it should generally show an increasing trend.
Since the kinetic energy of an object is given by the formula KE = (1/2)mv^2, where m is the mass of the object and v is its velocity, we need to calculate the velocity of the ball when it is dropped from each height in order to determine its kinetic energy.
Let's start with the ball dropped from a height of 1 meter. The potential energy of the ball at this height can be calculated using the formula PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height. Given that m = 10 kg, g = 9.8 m/s^2, and h = 1 m, we can calculate the potential energy:
PE = mgh
PE = (10 kg)(9.8 m/s^2)(1 m)
PE = 98 J
Since the potential energy is converted entirely into kinetic energy, we can equate the two:
KE = PE
(1/2)mv^2 = 98 J
Simplifying the equation, we have:
(v^2)/2 = 98 J/(10 kg)
v^2 = (2)(98 J)/(10 kg)
v^2 = 19.6 m^2/s^2
v = sqrt(19.6) m/s
v ≈ 4.427 m/s
Now let's do the same calculation for the ball dropped from a height of 2 meters. Using the same formula for potential energy, we have:
PE = mgh
PE = (10 kg)(9.8 m/s^2)(2 m)
PE = 196 J
Equating potential energy and kinetic energy, we have:
(1/2)mv^2 = 196 J
v^2 = (2)(196 J)/(10 kg)
v^2 = 39.2 m^2/s^2
v = sqrt(39.2) m/s
v ≈ 6.259 m/s
Finally, let's calculate the velocity for the ball dropped from a height of 3 meters:
PE = mgh
PE = (10 kg)(9.8 m/s^2)(3 m)
PE = 294 J
Equating potential and kinetic energy:
(1/2)mv^2 = 294 J
v^2 = (2)(294 J)/(10 kg)
v^2 = 58.8 m^2/s^2
v = sqrt(58.8) m/s
v ≈ 7.672 m/s
In summary, the velocities of the ball when dropped from heights of 1 meter, 2 meters, and 3 meters are approximately 4.427 m/s, 6.259 m/s, and 7.672 m/s respectively.
Now, we can plot these velocities on a graph against their corresponding heights. The correct graph will have the heights on the x-axis and the velocities on the y-axis.
Graph (A):
- x-axis: Height (m)
- y-axis: Velocity (m/s)
- Three points plotted: (1, 4.427), (2, 6.259), (3, 7.672)
Therefore, graph (A) correctly shows the ball's kinetic energy when dropped from each height.
To determine the correct graph showing the ball's kinetic energy when dropped from different heights, we need to understand the concept of kinetic energy.
Kinetic energy (KE) is the energy possessed by an object due to its motion. It depends on two factors: the mass of the object (m) and its velocity (v). Mathematically, it is expressed as KE = 0.5 * m * v^2.
In the given scenario, the mass (m) of the bowling ball is constant at 10 kilograms. However, the velocity (v) will vary depending on the height from which the ball is dropped.
To find the velocity, we can use the concept of gravitational potential energy (PE) and the law of conservation of energy. When an object is dropped from a certain height, its potential energy is converted into kinetic energy as it falls freely under the influence of gravity.
The potential energy at a given height is given by PE = m * g * h, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Since the potential energy is converted into kinetic energy, we can equate the two equations:
PE = KE
m * g * h = 0.5 * m * v^2
Now we can solve for the velocity (v):
v = sqrt(2 * g * h)
Using this equation, we can calculate the velocity of the ball when dropped from each height:
- From a height of 1 meter: v1 = sqrt(2 * 9.8 * 1) ≈ 4.43 m/s
- From a height of 2 meters: v2 = sqrt(2 * 9.8 * 2) ≈ 6.26 m/s
- From a height of 3 meters: v3 = sqrt(2 * 9.8 * 3) ≈ 8.66 m/s
Now that we have the velocities, we can calculate the kinetic energy using the equation KE = 0.5 * m * v^2:
- KE1 = 0.5 * 10 * (4.43)^2 ≈ 97.87 Joules
- KE2 = 0.5 * 10 * (6.26)^2 ≈ 196.57 Joules
- KE3 = 0.5 * 10 * (8.66)^2 ≈ 375.89 Joules
Based on these calculations, the correct graph should show the following values for the kinetic energy:
Height (m) | Kinetic Energy (J)
-----------------------------------
1 | 97.87
2 | 196.57
3 | 375.89
Please note that the graph should have height on the x-axis and kinetic energy on the y-axis.