Weight force is given by:

We only need the equations for weight and normal force to solve this problem. First the equation for weight (force of gravity):

Now the normal force, which is a function of weight:

If you're unsure why we used cosine, think about the situation practically. As the hill gets flatter (angle decreased), the normal force grows, and cosine is the function that gets larger as the angle gets smaller.

From the problem statement we know that the ratio of weight to normal force is 3:

We will start this problem by using the following kinematics equation:

Note that each variable is oriented in the direction of the slope of the hill. We will find the distance traveled along the hill, and then convert that to a height using the slope of the hill.

We already know the value of each variable, so we can find the distance traveled:

Now we can use the slope of the hill and the sine function to determine the height dropped:

We know these values, so let's plug them in:

By constructing a force diagram, it can be seen that:

Right when the box starts to move, these will be equal

Solving for

Plugging in values:

Concerning the astronaut:

The force of gravity will be essentially constant and downward. Thus, in order for acceleration of the astronaut to increase, the normal force must increase.

In this problem the  force acts in the downward direction. The box's weight also acts in the downward direction. We can calculate weight of the box using the equation .  

Normal force acts in the opposition to the weight of the box. We can calculate normal force by adding up the downward forces to find the counteracting force on the box.  

Forces downward:

Forces upward: 

Forces downward are equal to the forces upward. Therefore, 

The correct answer is . There are 2 forces acting on the box: tension force () and normal force (). First we must calculate the force of the box:

The equation can be manipulated to solve for acceleration:

Weight is the force that gravity exerts on an object. Weight is determined using the equation 

Where  is the mass of the object and  is the gravitational constant on the given planet.

On Earth, , so the weight of the car on Earth is:

Weight is the force that gravity exerts on an object. Weight is determined using the equation 

Where  is the mass of the object and  is the gravitational constant on the given planet.

On Mars, , so the weight of the car on Mars is:

To begin solving our problem, we will create a free body diagram, labeling all our forces and the direction in which they are acting. 

The normal force, , is the component of force perpendicular to the surface of contact. Weight, , is a force that gravity exerts on an object, acting in the same direction as gravity. To find the normal force, we will begin with Newton's 2nd law:

Where  is the sum of all forces in the same direction,  is mass, and  is the acceleration in the direction of the forces.

Since our object is at rest,  and Newton's 2nd law becomes:

Assuming  is in the positive direction, the sum of our forces in that direction is

Substituting in the definition of weight, , and solving the expression for  gives us 

Since  and