If A Stone Is Thrown Vertically Downward Into A Well With A Speed Of $10 , \text{m/s}$, How Long Will It Take To Reach The Water Surface 60 Meters Below?

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Introduction

When a stone is thrown vertically downward into a well, it experiences a constant downward acceleration due to gravity. This acceleration causes the stone to accelerate at a rate of $9.8 , \text{m/s}^2$, which is the acceleration due to gravity on Earth. In this article, we will calculate the time it takes for the stone to reach the water surface 60 meters below, given an initial speed of $10 , \text{m/s}$.

Understanding the Motion

When the stone is thrown vertically downward, it experiences a downward acceleration due to gravity. This acceleration causes the stone to accelerate at a rate of $9.8 , \text{m/s}^2$. The initial speed of the stone is $10 , \text{m/s}$, which is the speed at which it is thrown downward.

Equations of Motion

To calculate the time it takes for the stone to reach the water surface, we need to use the equations of motion. The equation of motion that relates the initial speed, acceleration, and displacement is:

s=ut+12at2s = ut + \frac{1}{2}at^2

where $s$ is the displacement, $u$ is the initial speed, $t$ is the time, and $a$ is the acceleration.

Calculating the Time

We are given that the displacement $s$ is 60 meters, the initial speed $u$ is $10 , \text{m/s}$, and the acceleration $a$ is $9.8 , \text{m/s}^2$. We need to calculate the time $t$ it takes for the stone to reach the water surface.

Substituting the values into the equation of motion, we get:

60=10t+12×9.8×t260 = 10t + \frac{1}{2} \times 9.8 \times t^2

Simplifying the equation, we get:

60=10t+4.9t260 = 10t + 4.9t^2

Rearranging the equation to form a quadratic equation, we get:

4.9t2+10t−60=04.9t^2 + 10t - 60 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a$ is the coefficient of the $t^2$ term, $b$ is the coefficient of the $t$ term, and $c$ is the constant term.

Substituting the values into the quadratic formula, we get:

t=−10±102−4×4.9×(−60)2×4.9t = \frac{-10 \pm \sqrt{10^2 - 4 \times 4.9 \times (-60)}}{2 \times 4.9}

Simplifying the equation, we get:

t=−10±100+11769.8t = \frac{-10 \pm \sqrt{100 + 1176}}{9.8}

t=−10±12769.8t = \frac{-10 \pm \sqrt{1276}}{9.8}

t=−10±35.79.8t = \frac{-10 \pm 35.7}{9.8}

Calculating the Time

We have two possible values for the time $t$:

t1=−10+35.79.8t_1 = \frac{-10 + 35.7}{9.8}

t2=−10−35.79.8t_2 = \frac{-10 - 35.7}{9.8}

Simplifying the equations, we get:

t1=25.79.8t_1 = \frac{25.7}{9.8}

t2=−45.79.8t_2 = \frac{-45.7}{9.8}

t1=2.63 st_1 = 2.63 \, \text{s}

t2=−4.67 st_2 = -4.67 \, \text{s}

Since time cannot be negative, we discard the negative value $t_2$.

Conclusion

The time it takes for the stone to reach the water surface 60 meters below is $2.63 , \text{s}$. This is the time it takes for the stone to accelerate downward due to gravity and reach the water surface.

Understanding the Physics

The motion of the stone is an example of uniformly accelerated motion. The stone experiences a constant downward acceleration due to gravity, which causes it to accelerate at a rate of $9.8 , \text{m/s}^2$. The initial speed of the stone is $10 , \text{m/s}$, which is the speed at which it is thrown downward.

Real-World Applications

The concept of uniformly accelerated motion is used in many real-world applications, such as:

  • Projectile motion: The motion of a projectile, such as a thrown ball or a rocket, is an example of uniformly accelerated motion.
  • Motion of objects under gravity: The motion of objects under gravity, such as a falling object or a rolling ball, is an example of uniformly accelerated motion.
  • Motion of vehicles: The motion of vehicles, such as cars or trains, is an example of uniformly accelerated motion.

Conclusion

In conclusion, the time it takes for a stone to reach the water surface 60 meters below is $2.63 , \text{s}$. This is the time it takes for the stone to accelerate downward due to gravity and reach the water surface. The concept of uniformly accelerated motion is used in many real-world applications, such as projectile motion, motion of objects under gravity, and motion of vehicles.

Introduction

In our previous article, we calculated the time it takes for a stone to reach the water surface 60 meters below, given an initial speed of $10 , \text{m/s}$. In this article, we will answer some frequently asked questions related to the motion of the stone.

Q: What is the acceleration of the stone?

A: The acceleration of the stone is $9.8 , \text{m/s}^2$, which is the acceleration due to gravity on Earth.

Q: What is the initial speed of the stone?

A: The initial speed of the stone is $10 , \text{m/s}$, which is the speed at which it is thrown downward.

Q: What is the displacement of the stone?

A: The displacement of the stone is 60 meters, which is the distance from the top of the well to the water surface.

Q: What is the time it takes for the stone to reach the water surface?

A: The time it takes for the stone to reach the water surface is $2.63 , \text{s}$.

Q: What is the equation of motion that relates the initial speed, acceleration, and displacement?

A: The equation of motion that relates the initial speed, acceleration, and displacement is:

s=ut+12at2s = ut + \frac{1}{2}at^2

where $s$ is the displacement, $u$ is the initial speed, $t$ is the time, and $a$ is the acceleration.

Q: How do you calculate the time it takes for the stone to reach the water surface?

A: To calculate the time it takes for the stone to reach the water surface, you need to use the equation of motion and substitute the values of the displacement, initial speed, and acceleration.

Q: What is the significance of the acceleration due to gravity?

A: The acceleration due to gravity is the acceleration that an object experiences when it is under the influence of gravity. In this case, the acceleration due to gravity is $9.8 , \text{m/s}^2$, which is the acceleration that the stone experiences as it falls downward.

Q: What is the difference between uniformly accelerated motion and uniformly decelerated motion?

A: Uniformly accelerated motion is motion that is accelerated at a constant rate, while uniformly decelerated motion is motion that is decelerated at a constant rate. In this case, the stone experiences uniformly accelerated motion as it falls downward.

Q: What are some real-world applications of uniformly accelerated motion?

A: Some real-world applications of uniformly accelerated motion include:

  • Projectile motion: The motion of a projectile, such as a thrown ball or a rocket, is an example of uniformly accelerated motion.
  • Motion of objects under gravity: The motion of objects under gravity, such as a falling object or a rolling ball, is an example of uniformly accelerated motion.
  • Motion of vehicles: The motion of vehicles, such as cars or trains, is an example of uniformly accelerated motion.

Conclusion

In conclusion, the time it takes for a stone to reach the water surface 60 meters below is $2.63 , \text{s}$. This is the time it takes for the stone to accelerate downward due to gravity and reach the water surface. The concept of uniformly accelerated motion is used in many real-world applications, such as projectile motion, motion of objects under gravity, and motion of vehicles.

Frequently Asked Questions

  • Q: What is the acceleration of the stone? A: The acceleration of the stone is $9.8 , \text{m/s}^2$.
  • Q: What is the initial speed of the stone? A: The initial speed of the stone is $10 , \text{m/s}$.
  • Q: What is the displacement of the stone? A: The displacement of the stone is 60 meters.
  • Q: What is the time it takes for the stone to reach the water surface? A: The time it takes for the stone to reach the water surface is $2.63 , \text{s}$.

Glossary

  • Acceleration: The rate of change of velocity of an object.
  • Displacement: The distance between the initial and final positions of an object.
  • Equation of motion: A mathematical equation that relates the initial speed, acceleration, and displacement of an object.
  • Gravity: The force that attracts objects towards each other.
  • Uniformly accelerated motion: Motion that is accelerated at a constant rate.