Derivation of Acceleration

Oct 13, 2025 by miodeya.com 123 views

Introduction

Imagine yourself in a car at the top of a roller-coaster ride, feeling the rush of adrenaline as you prepare to embark on a thrilling journey. The distance, $d$ , is determined by the equation $d = 144 - 16t$ , where $t$ is the number of seconds. In this article, we will delve into the world of mathematics to understand the motion of Greg's car on the roller-coaster ride. We will explore the concept of motion, velocity, and acceleration, and determine the interval of time for which Greg's car is moving.

Motion and Velocity

Motion is the change in position of an object over time. In the context of the roller-coaster ride, the distance, $d$ , is a function of time, $t$ . The velocity of the car is the rate of change of distance with respect to time. Mathematically, velocity is represented as the derivative of distance with respect to time, denoted as $\frac{dd}{dt}$ .

To find the velocity of Greg's car, we need to differentiate the distance equation with respect to time. Using the power rule of differentiation, we get:

v = \frac{dd}{dt} = -16

The negative sign indicates that the velocity is in the opposite direction to the positive direction of the distance axis. This means that the car is moving downwards.

Acceleration

Acceleration is the rate of change of velocity with respect to time. Mathematically, acceleration is represented as the derivative of velocity with respect to time, denoted as $\frac{dv}{dt}$ .

To find the acceleration of Greg's car, we need to differentiate the velocity equation with respect to time. Since the velocity is constant, the acceleration is zero.

Interval of Time

Now that we have understood the motion of Greg's car, we need to determine the interval of time for which the car is moving. To do this, we need to find the time interval for which the distance is positive.

The distance equation is given by $d = 144 - 16t$ . We need to find the values of $t$ for which $d > 0$ . This means that we need to solve the inequality $144 - 16t > 0$ .

Solving the inequality, we get:

t < \frac{144}{16}

t < 9

This means that the car is moving for $t < 9$ seconds.

Conclusion

In conclusion, we have understood the motion of Greg's car on the roller-coaster ride. We have found the velocity and acceleration of the car, and determined the interval of time for which the car is moving. The car is moving for $t < 9$ seconds, and the velocity is constant at $-16$ units per second.

Mathematical Derivations

Derivation of Velocity

To find the velocity of Greg's car, we need to differentiate the distance equation with respect to time. Using the power rule of differentiation, we get:

v = \frac{dd}{dt} = -16

Derivation of Acceleration

To find the acceleration of Greg's car, we need to differentiate the velocity equation with respect to time. Since the velocity is constant, the acceleration is zero.

Solution of Inequality

To find the interval of time for which the car is moving, we need to solve the inequality $144 - 16t > 0$ . Solving the inequality, we get:

t < \frac{144}{16}

t < 9

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Physics for Scientists and Engineers, 3rd edition, Paul A. Tipler

Glossary

Motion: The change in position of an object over time.
Velocity: The rate of change of distance with respect to time.
Acceleration: The rate of change of velocity with respect to time.
Distance: The length of the path traveled by an object.
Time: A measure of the duration of an event or process.
Frequently Asked Questions: Understanding Motion on a Roller-Coaster ====================================================================

Q: What is the equation for the distance of Greg's car on the roller-coaster ride?

A: The equation for the distance of Greg's car on the roller-coaster ride is given by $d = 144 - 16t$ , where $t$ is the number of seconds.

Q: What is the velocity of Greg's car on the roller-coaster ride?

A: The velocity of Greg's car on the roller-coaster ride is constant at $-16$ units per second. This means that the car is moving downwards.

Q: What is the acceleration of Greg's car on the roller-coaster ride?

A: The acceleration of Greg's car on the roller-coaster ride is zero. This means that the velocity of the car is constant.

Q: For which interval of time is Greg's car moving on the roller-coaster ride?

A: Greg's car is moving for $t < 9$ seconds. This means that the car is moving for the entire duration of the ride, until it reaches the bottom of the roller-coaster.

Q: What is the significance of the negative sign in the velocity equation?

A: The negative sign in the velocity equation indicates that the velocity is in the opposite direction to the positive direction of the distance axis. This means that the car is moving downwards.

Q: How can we determine the interval of time for which the car is moving?

A: We can determine the interval of time for which the car is moving by solving the inequality $144 - 16t > 0$ . Solving the inequality, we get $t < 9$ .

Q: What is the relationship between distance, velocity, and acceleration?

A: The distance, velocity, and acceleration are related by the following equations:

Distance: $d = 144 - 16t$
Velocity: $v = \frac{dd}{dt} = -16$
Acceleration: $a = \frac{dv}{dt} = 0$

Q: How can we use the equations to determine the motion of the car?

A: We can use the equations to determine the motion of the car by substituting the values of $t$ into the distance equation. This will give us the distance of the car at any given time.

Q: What is the significance of the roller-coaster ride in this context?

A: The roller-coaster ride is a real-world example of motion, where the car is moving in a curved path. The equations we derived can be used to describe the motion of the car on the roller-coaster ride.

Q: How can we apply the concepts of motion to real-world situations?

A: We can apply the concepts of motion to real-world situations by using the equations to describe the motion of objects in different contexts. For example, we can use the equations to describe the motion of a ball thrown upwards, or the motion of a car on a straight road.

Q: What are some common applications of the concepts of motion?

A: Some common applications of the concepts of motion include:

Physics: Motion is a fundamental concept in physics, and is used to describe the behavior of objects in different contexts.
Engineering: Motion is used in engineering to design and optimize systems, such as roller-coasters and cars.
Sports: Motion is used in sports to analyze and improve performance, such as in track and field events.
Video games: Motion is used in video games to create realistic and engaging experiences.

Q: What are some common misconceptions about motion?

A: Some common misconceptions about motion include:

Motion is only about speed: While speed is an important aspect of motion, it is not the only factor. Acceleration and distance are also important.
Motion is only about objects: While objects are the primary focus of motion, motion can also be applied to abstract concepts, such as energy and momentum.
Motion is only about the past: While the past is an important aspect of motion, it is not the only factor. Motion can also be used to predict the future behavior of objects.