# Applications of Newton's Laws

• Shadowness
In summary, the conversation is about a person seeking help with a physics problem involving a frictionless track and a cart's movement. The question asks for the minimum time needed for the cart to slide from the top to the end of the track, and the hint suggests using derivatives to find the solution. The person is unsure of how to approach the problem and needs guidance.
Shadowness
Hello all,

I never really had a great deal of trouble with physics until just recently and this place looks like a great place to learn. I am having problems with the following question:

A frictionless track is to be built as shown, with L=3.20 m and H=2.90 m. In order to get the cart to slide from the top to the end of the track in the minimum time, how long should the distance D be? Assume that the speed of the cart on the horizontal surface is the same as at the bottom of the ramp.
Hint: A minimum or maximum value can be found using derivatives.
http://www.shadowsillusion.com/images/ramp2.gif

How would I go about in setting this problem up? I am just clueless one where to start. I believe that I have to set up some type of formula, take the derivative of it and solve for D?

Thank you for your time and help.

Last edited:
Some hints:

No friction means energy is conserved.
Acceleration for frictionless inclines is constant
Average velocity is easy to calculate when acceleration is constant
Average velocity is distance divided by time.
Total path length is a fairly easy trig problem

Tell us what you think needs to be done

Hi there,

Newton's Laws of Motion are fundamental principles in physics that explain how objects move and interact with each other. They have many practical applications in our daily lives, including in the design and construction of structures and machines, as well as in understanding the motion of celestial bodies.

In the case of the problem you mentioned, Newton's Laws can be used to determine the minimum time it takes for a cart to slide down a frictionless track. The first law, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. This means that once the cart is set in motion, it will continue to move down the track without any external forces acting on it.

The second law, also known as the law of acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the larger the force pushing the cart down the track, the faster it will accelerate. In this case, the force pushing the cart down the track is its weight, which is equal to its mass multiplied by the acceleration due to gravity.

The third law, also known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This means that the force of the track pushing up on the cart is equal and opposite to the force of the cart pushing down on the track.

To solve the problem, we can use the equation F=ma, where F is the force, m is the mass, and a is the acceleration. We can also use the equation for the acceleration due to gravity, a=g, where g is approximately 9.8 m/s^2. We can set up the problem as follows:

F=ma
mg=ma
g=a

Using the third law, we know that the force pushing the cart down the track is equal to its weight, which is mg. We also know that the acceleration of the cart is equal to the acceleration due to gravity, which is g. Therefore, we can plug in these values into the equation for acceleration and solve for g.

g=9.8 m/s^2

Next, we can use the equation for displacement, d=vt+1/2at^2, where d is the displacement, v is the initial velocity, t is the time, and a

## 1. What are Newton's Laws of Motion?

Newton's Laws of Motion are a set of three principles that describe the behavior of objects in motion. They were developed by Sir Isaac Newton in the 17th century and are considered the foundation of classical mechanics.

## 2. What is the first law of motion?

The first law, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will continue moving in a straight line at a constant speed, unless acted upon by an external force.

## 3. How is the second law of motion applied in real life?

The second law states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass. This law is applied in everyday situations such as driving a car, throwing a ball, or pushing a shopping cart.

## 4. What is the third law of motion?

The third law states that for every action, there is an equal and opposite reaction. This means that when an object exerts a force on another object, the second object will exert an equal force in the opposite direction.

## 5. How are Newton's Laws used in engineering and technology?

Engineers and technologists use Newton's Laws to design and create various machines and structures that rely on the principles of motion and force. For example, the design of a roller coaster or a bridge requires an understanding of these laws to ensure safety and functionality.

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