A Stone Is Thrown From The Top Of A Tall Cliff. Its Acceleration Is A Constant $-32 \frac{\text{ft}}{\text{sec}^2}$ (so $A(t)=-32$). Its Velocity After 2 Seconds Is $18 \frac{\text{ft}}{\text{sec}}$, And Its Height After 2

Introduction

When a stone is thrown from the top of a tall cliff, it experiences a constant downward acceleration due to the force of gravity. The acceleration is a fundamental concept in physics that describes the rate of change of velocity of an object. In this article, we will explore the physics behind the motion of the stone, including its acceleration, velocity, and height as a function of time.

The Physics of Acceleration

The acceleration of the stone is a constant $-32 \frac{\text{ft}}{\text{sec}^2}$, which means that the velocity of the stone is decreasing at a rate of 32 feet per second squared. This is because the force of gravity is acting downward on the stone, causing it to accelerate downward.

The Equation of Motion

The equation of motion for an object under constant acceleration is given by:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

where $s(t)$ is the position of the object at time $t$, $s_0$ is the initial position, $v_0$ is the initial velocity, and $a$ is the acceleration.

The Velocity of the Stone

The velocity of the stone after 2 seconds is given as $18 \frac{\text{ft}}{\text{sec}}$. We can use the equation of motion to find the velocity of the stone at any time $t$:

$$v(t) = v_0 + at$$

Substituting the values of $v_0 = 18 \frac{\text{ft}}{\text{sec}}$ and $a = -32 \frac{\text{ft}}{\text{sec}^2}$, we get:

$$v(t) = 18 - 32t$$

The Height of the Stone

The height of the stone after 2 seconds is not given, but we can use the equation of motion to find it. We know that the initial height is $s_0 = 0$ (since the stone is thrown from the top of the cliff), and the initial velocity is $v_0 = 18 \frac{\text{ft}}{\text{sec}}$. We can substitute these values into the equation of motion:

$$s(t) = 0 + 18t - 16t^2$$

Solving for the Height

To find the height of the stone after 2 seconds, we can substitute $t = 2$ into the equation:

$$s(2) = 0 + 18(2) - 16(2)^2$$

$$s(2) = 36 - 64$$

$$s(2) = -28$$

So, the height of the stone after 2 seconds is $-28$ feet.

Conclusion

In conclusion, the motion of the stone thrown from the top of a tall cliff can be described using the equation of motion. We can use this equation to find the velocity and height of the stone at any time $t$. The acceleration of the stone is a constant $-32 \frac{\text{ft}}{\text{sec}^2}$, which means that the velocity of the stone is decreasing at a rate of 32 feet per second squared.

The Physics of Projectile Motion

Projectile motion is a type of motion that occurs when an object is thrown or projected into the air. The motion of the stone thrown from the top of the cliff is an example of projectile motion. The key characteristics of projectile motion are:

• Horizontal motion: The stone moves horizontally at a constant velocity.
• Vertical motion: The stone moves vertically under the influence of gravity.
• Acceleration: The stone experiences a constant downward acceleration due to gravity.

The Trajectory of the Stone

The trajectory of the stone is a parabola that opens downward. The stone follows a curved path under the influence of gravity, with the highest point of the trajectory occurring at the peak of the parabola.

The Time of Flight

The time of flight of the stone is the time it takes for the stone to reach the ground. We can find the time of flight by using the equation of motion:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

Substituting the values of $s_0 = 0$, $v_0 = 18 \frac{\text{ft}}{\text{sec}}$, and $a = -32 \frac{\text{ft}}{\text{sec}^2}$, we get:

$$s(t) = 0 + 18t - 16t^2$$

To find the time of flight, we can set $s(t) = 0$ and solve for $t$:

$$0 = 18t - 16t^2$$

$$16t^2 - 18t = 0$$

$$t(16t - 18) = 0$$

$$t = 0 \text{ or } t = \frac{18}{16}$$

Since $t = 0$ is the initial time, we can discard it and find the time of flight:

$$t = \frac{18}{16} = 1.125 \text{ seconds}$$

So, the time of flight of the stone is 1.125 seconds.

The Range of the Stone

The range of the stone is the horizontal distance it travels before hitting the ground. We can find the range by using the equation of motion:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

Substituting the values of $s_0 = 0$, $v_0 = 18 \frac{\text{ft}}{\text{sec}}$, and $a = -32 \frac{\text{ft}}{\text{sec}^2}$, we get:

$$s(t) = 0 + 18t - 16t^2$$

To find the range, we can set $t = 1.125$ seconds (the time of flight) and solve for $s(t)$:

$$s(1.125) = 0 + 18(1.125) - 16(1.125)^2$$

$$s(1.125) = 20.25 - 16(1.265625)$$

$$s(1.125) = 20.25 - 20.25$$

$$s(1.125) = 0$$

So, the range of the stone is 0 feet.

Conclusion

In conclusion, the motion of the stone thrown from the top of a tall cliff can be described using the equation of motion. We can use this equation to find the velocity, height, and time of flight of the stone. The acceleration of the stone is a constant $-32 \frac{\text{ft}}{\text{sec}^2}$, which means that the velocity of the stone is decreasing at a rate of 32 feet per second squared.

The Physics of Projectile Motion: A Summary

Projectile motion is a type of motion that occurs when an object is thrown or projected into the air. The key characteristics of projectile motion are:

• Horizontal motion: The object moves horizontally at a constant velocity.
• Vertical motion: The object moves vertically under the influence of gravity.
• Acceleration: The object experiences a constant downward acceleration due to gravity.

The trajectory of the object is a parabola that opens downward, with the highest point of the trajectory occurring at the peak of the parabola. The time of flight of the object is the time it takes for the object to reach the ground, and the range of the object is the horizontal distance it travels before hitting the ground.

The Applications of Projectile Motion

Projectile motion has many practical applications in fields such as:

• Sports: Projectile motion is used in sports such as baseball, basketball, and football to predict the trajectory of a thrown or kicked ball.
• Aerospace Engineering: Projectile motion is used in aerospace engineering to design and test the trajectory of spacecraft and missiles.
• Physics Education: Projectile motion is used in physics education to teach students about the fundamental principles of motion and gravity.

Conclusion

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Q&A: Projectile Motion and the Stone Thrown from the Cliff

Q: What is projectile motion?

A: Projectile motion is a type of motion that occurs when an object is thrown or projected into the air. The object moves horizontally at a constant velocity and vertically under the influence of gravity.

Q: What are the key characteristics of projectile motion?

A: The key characteristics of projectile motion are:

• Horizontal motion: The object moves horizontally at a constant velocity.
• Vertical motion: The object moves vertically under the influence of gravity.
• Acceleration: The object experiences a constant downward acceleration due to gravity.

Q: What is the trajectory of a projectile?

A: The trajectory of a projectile is a parabola that opens downward, with the highest point of the trajectory occurring at the peak of the parabola.

Q: What is the time of flight of a projectile?

A: The time of flight of a projectile is the time it takes for the projectile to reach the ground.

Q: How can we find the time of flight of a projectile?

A: We can find the time of flight of a projectile by using the equation of motion:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

where $s(t)$ is the position of the projectile at time $t$, $s_0$ is the initial position, $v_0$ is the initial velocity, and $a$ is the acceleration.

Q: What is the range of a projectile?

A: The range of a projectile is the horizontal distance it travels before hitting the ground.

Q: How can we find the range of a projectile?

A: We can find the range of a projectile by using the equation of motion:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

where $s(t)$ is the position of the projectile at time $t$, $s_0$ is the initial position, $v_0$ is the initial velocity, and $a$ is the acceleration.

Q: What is the acceleration of a projectile?

A: The acceleration of a projectile is a constant downward acceleration due to gravity, which is $-32 \frac{\text{ft}}{\text{sec}^2}$.

Q: How can we find the velocity of a projectile?

A: We can find the velocity of a projectile by using the equation of motion:

$$v(t) = v_0 + at$$

where $v(t)$ is the velocity of the projectile at time $t$, $v_0$ is the initial velocity, and $a$ is the acceleration.

Q: What is the height of a projectile?

A: The height of a projectile is the vertical distance it travels before hitting the ground.

Q: How can we find the height of a projectile?

A: We can find the height of a projectile by using the equation of motion:

$$s(t) = s_0 + v_0t + \frac{1}{2}at^2$$

where $s(t)$ is the position of the projectile at time $t$, $s_0$ is the initial position, $v_0$ is the initial velocity, and $a$ is the acceleration.

Q: What are some practical applications of projectile motion?

A: Some practical applications of projectile motion include:

• Sports: Projectile motion is used in sports such as baseball, basketball, and football to predict the trajectory of a thrown or kicked ball.
• Aerospace Engineering: Projectile motion is used in aerospace engineering to design and test the trajectory of spacecraft and missiles.
• Physics Education: Projectile motion is used in physics education to teach students about the fundamental principles of motion and gravity.

Conclusion

In conclusion, projectile motion is a type of motion that occurs when an object is thrown or projected into the air. The key characteristics of projectile motion are horizontal motion, vertical motion, and acceleration. We can use the equation of motion to find the time of flight, range, velocity, and height of a projectile. Projectile motion has many practical applications in fields such as sports, aerospace engineering, and physics education.