For this problem, we are looking at the net force on the bone. Since both dogs are pulling with the same force, we can say the magnitude of the force of dog 1 equals the magnitude of the force of dog 2, but in opposite directions. Mathematically, \(\displaystyle F_1=-F_2\).

To find the net force on the bone, we add the individual forces together.

\(\displaystyle F_{net}=F_1+F_2\)

We can substitute our forces into the equation for net force.

\(\displaystyle F_{net}=(-F_2)+F_2\)

\(\displaystyle F_{net}=0N\)

The net force is zero. Each dog pulls with equal force in opposite directions, allowing the total force to cancel out.

According to Newton's second law, \(\displaystyle F_{net}=ma\). If the net force is zero, then the acceleration of the bone must also be zero.

\(\displaystyle 0N=ma\)

\(\displaystyle a=0\frac{m}{s^2}\)

If the drop is at rest, then that means that the net forces acting upon it are equal to zero. There are two forces acting on the drop: the force due to gravity and the frictional force of the paint on the wall.

Mathematically, we can set up an equation for the net force:

\(\displaystyle F_{net}=F_f-F_G\)

\(\displaystyle 0=F_f-F_G\)

\(\displaystyle -F_G=F_f\)

The forces of friction and gravity are going to be equal and opposite, causing the drop to remain still on the wall.

If two forces act on a single object, then the net force on the object is equal to the sum of the forces acting on it.

\(\displaystyle F_{net}=F_1+F_2+...\)

Forces are vector quantities, however. This means that all forces have a magnitude and a direction of action. When adding forces, we must take their directions into account. Directions are broken into horizontal and vertical portions for vector addition, and can be positive or negative based on the direction along the axis.

For a net force to be zero, one force must be positive and the other must be negative along a given axis. If the forces are of equal magnitude, then they must be acting in exactly opposite directions in order to cancel each other.

\(\displaystyle F_1=-F_2\)

\(\displaystyle F_{net}=F_1+F_2=(-F_2)+F_2=0\)

Kinetic force is not a technically correct property. Kinetic energy can be used to generate force, but is not a force in itself. Forces require non-zero acceleration, meaning that they only exist when there is a changein velocity. A change in kinetic energy can indicate a change in velocity, leading to a non-zero force value.

Normal force is the force of a surface upon an object, frequently countering gravity. Friction force is the resistance between an object and a surface, and acts opposite the direction of the object's motion. Buoyant force is the upward force of a fluid on a submerged object. Drag, or air resistance, is a special form of friction force for an object moving through a fluid.

First, realize that the force that the lighter child exerts on heavier child is equal and opposite to the force the heavier child exerts on the lighter child, as per Newton's third law.

\(\displaystyle F_1=-F_2\)

Using Newton's second law, we can re-write this equation.

\(\displaystyle m_1a_1=-m_2a_2\)

The question tells us that \(\displaystyle m_1=2m_2\), making \(\displaystyle m_1\) the heavier child and \(\displaystyle m_2\) the lighter child. We can use this in our equation as well.

\(\displaystyle 2m_2a_1=m_2a_2\)

We are looking for the ratio of \(\displaystyle a_2\) to \(\displaystyle a_1\), so we need to rearrange the equation.

First, the masses cancel out.

\(\displaystyle 2a_1=a_2\)

Then, divide both sides by \(\displaystyle a_1\).

\(\displaystyle 2=\frac{a_2}{a_1}\)

The ratio of  \(\displaystyle a_2\) to \(\displaystyle a_1\) is \(\displaystyle 2:1\).

The relationship between force and acceleration is \(\displaystyle F_{net}=ma\).

Since the crate has a constant velocity, it has no acceleration.

\(\displaystyle a=\frac{\Delta v}{t}=\frac{0}{t}=0\frac{m}{s^2}\)

If there is zero acceleration, that means there is no net force on the object, or \(\displaystyle F_{net}=0N\).

\(\displaystyle F_{net}=ma=m(0\frac{m}{s^2})=0N\)

Newton's second law states that \(\displaystyle F=ma\). We know the mass, but we need to calculate the acceleration.

Acceleration is the change in velocity per unit time.

\(\displaystyle a=\frac{v_2-v_1}{t}\)

Since the velocity does not change from one moment to the next, then there must be no net acceleration on the object.

\(\displaystyle a=\frac{3\frac{m}{s}-3\frac{m}{s}}{t}=\frac{0\frac{m}{s}}{t}\)

\(\displaystyle a=0\frac{m}{s^2}\)

Returning to Newton's second law, we can see that if there is no acceleration, then there is no net force.

\(\displaystyle F=ma\)

\(\displaystyle F=7kg*0\frac{m}{s^2}\)

\(\displaystyle F=0N\)

The equation for a force is:

\(\displaystyle F=ma\)

We can write this equation in terms of each object:

\(\displaystyle F_1=m_1a_1\)

\(\displaystyle F_2=m_2a_2\)

We know that the force applied to each object will be equal, so we can set these equations equal to each other.

\(\displaystyle m_1a_1=m_2a_2\)

We know that the second object is twice the mass of the first.

\(\displaystyle m_1a_1=2m_1a_2\)

We can cancel out the mass from each side, leaving a relationship between the two accelerations.

\(\displaystyle a_1=2a_2\)

The acceleration on the first mass is twice the acceleration on the second; thus, the acceleration of the lighter mass is greater.

The formula for force is Newton's second law:

\(\displaystyle F=ma\)

We are told in the question to use \(\displaystyle m\) for the mass and \(\displaystyle N\) for the force.

\(\displaystyle N=ma\)

Now we can isolate the acceleration.

\(\displaystyle \frac{N}{m}=a\)

This also makes sense from a units perspective. Units for force are Newtons, which can be written as:

\(\displaystyle 1N=1\frac{kg\cdot m}{s^2}\)

In our equation, we can see that Newtons are divided by mass:

\(\displaystyle \frac{N}{m}=\frac{\frac{kg\cdot m}{s^2}}{kg}=\frac{m}{s^2}\)

This would result in the units for acceleration.

If there are no forces acting upon the object, then there is no acceleration. If there is no acceleration, then the object will move with a constant velocity.

Mathematically, we can look at Newton's second law and the formula for acceleration.

\(\displaystyle F=ma\)

We know that the force is zero.

\(\displaystyle 0N=ma\)

Since we know that the mass cannot be zero, the acceleration must be zero.

\(\displaystyle 0N=m(0\frac{m}{s^2})\)

We can now use the formula for acceleration to see the effects on velocity.

\(\displaystyle a=\frac{v_2-v_1}{t}\)

We know that the acceleration is zero and that the time is ten seconds.

\(\displaystyle 0\frac{m}{s^2}=\frac{v_2-v_1}{10s}\)

In order for this to be true, the initial and final velocities must be equal.

\(\displaystyle v_2-v_1=0\)

\(\displaystyle v_1=v_2\)