Question

Write Newton's Second Law of Motion for three-dimensional motion with only the gravitational force (acting in the \(z\) -direction).

Short answer

Answer: The equations of motion for this case are: \(a_x = 0\), \(a_y = 0\), and \(a_z = -g\), where \(a_x\), \(a_y\), and \(a_z\) are the accelerations in the x, y, and z directions, respectively, and g is the acceleration due to gravity.

Step-by-step solution

Recalling Newton's Second Law of Motion

Newton's Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it is expressed as: $$ \vec{F} = m\vec{a} $$ where \(\vec{F}\) is the net force acting on an object, \(m\) is the mass of the object, and \(\vec{a}\) is the acceleration of the object.

Considering the three-dimensional motion

In three-dimensional motion, the net force and acceleration both have x, y, and z components. We can represent Newton's Second Law of Motion in component form as follows: $$ F_x = m a_x $$ $$ F_y = m a_y $$ $$ F_z = m a_z $$

Identifying the given conditions

In this problem, it is given that the only force acting on the object is the gravitational force and it is acting in the z-direction. Therefore, the force components in the x and y directions (\(F_x\) and \(F_y\)) are zero. The gravitational force acting in the z-direction is given by: $$ F_z = -mg $$ where \(g\) is the acceleration due to gravity and the negative sign indicates that the force is acting in the negative z-direction (downwards).

Applying the given conditions to Newton's Second Law

Now that we know the force components, we can apply the given condition to the component form of Newton's Second Law which we derived in step 2: $$ F_x = 0 = m a_x $$ $$ F_y = 0 = m a_y $$ $$ F_z = -mg = m a_z $$

Simplifying the equations

Finally, we can simplify these equations to get the equation of motion for this specific case: $$ a_x = 0 $$ $$ a_y = 0 $$ $$ a_z = -g $$ These equations represent Newton's Second Law of Motion for three-dimensional motion with only the gravitational force acting in the z-direction.