Deriving Motion Equation from Newton's Second Law

In summary, the problem involves an object being dropped from a height above a pool of water. The general expression for acceleration, ay=d2y/dt2, is used to generate a differential equation, y''(t) = -g. The next step is to integrate this equation, leading to the final equation of motion, y1=y2+vt - 1/2gt2. The approach to solving this problem involves finding the initial velocity and initial position, using them to determine the constants C and y2, and then substituting into the final equation.
  • #1
erok81
464
0

Homework Statement



An object is dropped from a height above a pool of water.

Start with Newton’s 2nd law, F = ma and use the general expression for acceleration, ay=d2y/dt2 to generate a differential equation

Homework Equations



F=ma

-mg=m d2y/dt2

The Attempt at a Solution



We had this problem in our problem solving session last week. My physics I class I never learned how to derive anything, so I am always on a quest to figure these out. This is one I cannot do. I can set up as far as I have. I see I can cancel the m's and then integrate both sides, but I cannot get to the correct answer.

The answer is a regular equation of motion - y1=y2+vt - 1/2gt2.

How does one approach this problem?
 
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  • #2
erok81 said:

Homework Statement



An object is dropped from a height above a pool of water.

Start with Newton’s 2nd law, F = ma and use the general expression for acceleration, ay=d2y/dt2 to generate a differential equation

Homework Equations



F=ma

-mg=m d2y/dt2

The Attempt at a Solution



We had this problem in our problem solving session last week. My physics I class I never learned how to derive anything, so I am always on a quest to figure these out. This is one I cannot do. I can set up as far as I have. I see I can cancel the m's and then integrate both sides, but I cannot get to the correct answer.

The answer is a regular equation of motion - y1=y2+vt - 1/2gt2.

How does one approach this problem?

You have the right equation to start.

y''(t) = -g

Here's the next step. Integrate once:

y'(t) = -gt + C

Use the velocity at t = 0 to figure out C, then integrate again, getting another constant you will need to figure out from the initial conditions.
 

1. How do you derive the motion equation from Newton's Second Law?

The motion equation, also known as the equation of motion or kinematic equation, can be derived from Newton's Second Law by using the formula F=ma, where F is the force applied on an object, m is the mass of the object, and a is the acceleration of the object. By rearranging the formula to solve for a, we get a=F/m. This value for acceleration can then be substituted into the kinematic equation a=(vf-vi)/t, where vf is the final velocity of the object, vi is the initial velocity, and t is the time interval. By plugging in the value for acceleration, we can then solve for the final velocity, giving us the motion equation vf=vi+at.

2. What is Newton's Second Law and how is it related to motion?

Newton's Second Law states that the acceleration of an object is directly proportional to the net force applied on the object, and inversely proportional to its mass. This means that the greater the force applied on an object, the greater its acceleration will be. This law is directly related to motion because it explains how forces act on objects and how those forces affect an object's motion.

3. Can the motion equation be used for any object?

Yes, the motion equation can be used for any object as long as the object is experiencing a net force and has a known mass and initial velocity. This equation is commonly used in physics and engineering to predict the motion of objects under the influence of various forces.

4. Is the motion equation the same as the equation of motion?

Yes, the motion equation and the equation of motion are different names for the same mathematical formula. This formula is used to calculate the final velocity of an object based on its initial velocity, acceleration, and time interval.

5. Can the motion equation be used to calculate the acceleration of an object?

Yes, the motion equation can be rearranged to solve for acceleration by using the formula a=(vf-vi)/t. This is often used when the acceleration is unknown and needs to be calculated based on the initial and final velocities of an object and the time interval.

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