A Tennis Ball Is Dropped From A Height. Assuming The Acceleration Due To Gravity Is $10 \, \text{m/s}^2$, Find:i.) The Time Elapsed Before It Hits The Ground.
Introduction
When a tennis ball is dropped from a height, it experiences a downward acceleration due to the force of gravity. This phenomenon is a fundamental concept in physics, and understanding it can help us predict the motion of objects under the influence of gravity. In this article, we will explore the physics of free fall and calculate the time elapsed before the tennis ball hits the ground.
The Physics of Free Fall
When an object is dropped from a height, it experiences a downward acceleration due to the force of gravity. The acceleration due to gravity is denoted by the symbol g and is approximately equal to 10 m/s^2 on the surface of the Earth. This acceleration is a result of the gravitational force exerted by the Earth on the object.
The Equation of Motion
To calculate the time elapsed before the tennis ball hits the ground, we need to use the equation of motion under constant acceleration. The equation is given by:
s = ut + (1/2)at^2
where s is the displacement of the object, u is the initial velocity, t is the time elapsed, and a is the acceleration due to gravity.
Initial Conditions
In this problem, the initial velocity of the tennis ball is zero, since it is dropped from rest. The displacement of the tennis ball is equal to the height from which it is dropped, which we will denote by h. The acceleration due to gravity is 10 m/s^2.
Solving the Equation of Motion
Substituting the initial conditions into the equation of motion, we get:
h = 0 + (1/2)(10)t^2
Simplifying the equation, we get:
h = 5t^2
Finding the Time Elapsed
To find the time elapsed before the tennis ball hits the ground, we need to solve for t. Since the displacement of the tennis ball is equal to the height from which it is dropped, we can set h equal to the height and solve for t.
h = 5t^2
h = 5(0)^2
h = 0
This is not possible, since the tennis ball is dropped from a height. Therefore, we need to use a different approach to solve for t.
Using the Quadratic Formula
The equation h = 5t^2 is a quadratic equation in t. We can solve for t using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 5, b = 0, and c = -h. Substituting these values into the quadratic formula, we get:
t = (0 ± √(0^2 - 4(5)(-h))) / 2(5)
Simplifying the equation, we get:
t = (0 ± √(20h)) / 10
Simplifying the Equation
Since the time elapsed cannot be negative, we can ignore the negative root and simplify the equation to:
t = √(20h) / 10
Finding the Time Elapsed
To find the time elapsed before the tennis ball hits the ground, we need to substitute the value of h into the equation. Let's assume that the height from which the tennis ball is dropped is 20 meters.
h = 20 m
Substituting this value into the equation, we get:
t = √(20(20)) / 10
t = √(400) / 10
t = 20 / 10
t = 2 s
Conclusion
In this article, we have explored the physics of free fall and calculated the time elapsed before a tennis ball hits the ground. We have used the equation of motion under constant acceleration to solve for t and have found that the time elapsed is approximately 2 seconds. This result is consistent with our intuitive understanding of the motion of objects under the influence of gravity.
Discussion
The calculation of the time elapsed before the tennis ball hits the ground is a classic problem in physics. It is a great example of how the equation of motion under constant acceleration can be used to solve real-world problems. The result of this calculation is consistent with our intuitive understanding of the motion of objects under the influence of gravity.
References
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
- Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
Further Reading
- For a more detailed discussion of the physics of free fall, see the article "Free Fall" by the Physics Classroom.
- For a more detailed discussion of the equation of motion under constant acceleration, see the article "Equation of Motion" by the Physics Classroom.
Introduction
In our previous article, we explored the physics of free fall and calculated the time elapsed before a tennis ball hits the ground. In this article, we will answer some of the most frequently asked questions about the physics of free fall.
Q: What is the acceleration due to gravity?
A: The acceleration due to gravity is the rate at which an object falls towards the ground. It is denoted by the symbol g and is approximately equal to 10 m/s^2 on the surface of the Earth.
Q: What is the equation of motion under constant acceleration?
A: The equation of motion under constant acceleration is given by:
s = ut + (1/2)at^2
where s is the displacement of the object, u is the initial velocity, t is the time elapsed, and a is the acceleration due to gravity.
Q: What is the initial velocity of an object dropped from rest?
A: The initial velocity of an object dropped from rest is zero. This is because the object is not moving initially, and therefore its velocity is zero.
Q: What is the displacement of an object dropped from a height?
A: The displacement of an object dropped from a height is equal to the height from which it is dropped. This is because the object falls from the initial height to the ground, and therefore its displacement is equal to the height.
Q: How do you calculate the time elapsed before an object hits the ground?
A: To calculate the time elapsed before an object hits the ground, you need to use the equation of motion under constant acceleration. You can substitute the values of the initial velocity, acceleration due to gravity, and displacement into the equation to solve for time.
Q: What is the time elapsed before a tennis ball hits the ground?
A: The time elapsed before a tennis ball hits the ground is approximately 2 seconds. This is because the tennis ball is dropped from a height of 20 meters, and the acceleration due to gravity is 10 m/s^2.
Q: What is the relationship between the time elapsed and the height from which an object is dropped?
A: The time elapsed before an object hits the ground is directly proportional to the square root of the height from which it is dropped. This is because the equation of motion under constant acceleration involves the square root of the displacement.
Q: Can you give an example of how to use the equation of motion under constant acceleration to solve a problem?
A: Yes, let's say you drop a ball from a height of 50 meters. You want to know how long it takes for the ball to hit the ground. You can use the equation of motion under constant acceleration to solve for time:
s = ut + (1/2)at^2
Substituting the values of the initial velocity (0), acceleration due to gravity (10 m/s^2), and displacement (50 m), you get:
50 = 0 + (1/2)(10)t^2
Simplifying the equation, you get:
50 = 5t^2
Solving for t, you get:
t = √(10) / √(5)
t ≈ 2.24 s
Therefore, it takes approximately 2.24 seconds for the ball to hit the ground.
Q: What are some real-world applications of the physics of free fall?
A: The physics of free fall has many real-world applications, including:
- Calculating the time elapsed before an object hits the ground
- Determining the height from which an object is dropped
- Understanding the motion of objects under the influence of gravity
- Designing safety features for buildings and bridges
- Developing models for predicting the motion of objects in space
Conclusion
In this article, we have answered some of the most frequently asked questions about the physics of free fall. We have discussed the equation of motion under constant acceleration, the initial velocity of an object dropped from rest, and the displacement of an object dropped from a height. We have also provided examples of how to use the equation of motion under constant acceleration to solve problems and discussed some real-world applications of the physics of free fall.
Discussion
The physics of free fall is a fundamental concept in physics that has many real-world applications. Understanding the motion of objects under the influence of gravity is essential for designing safety features for buildings and bridges, developing models for predicting the motion of objects in space, and calculating the time elapsed before an object hits the ground.
References
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
- Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
Further Reading
- For a more detailed discussion of the physics of free fall, see the article "Free Fall" by the Physics Classroom.
- For a more detailed discussion of the equation of motion under constant acceleration, see the article "Equation of Motion" by the Physics Classroom.