A space probe has two engines. Each generates the same amount of force when fired, and the directions of these forces can be independently adjusted. When the engines are fired simultaneously and each applies its force in the same direction, the probe, starting from rest, takes 48 s to travel a certain distance. How long does it take to travel the same distance, again starting from rest, if the engines are fired simultaneously and the forces that they apply to the probe are perpendicular?

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If the forces are perpendicular, the NET force is half way between the two engine exhaust directions and the magnitude is sqrt2 = 1.414 times the .

With the net force reduced by a factor 1/1.414 = 0.707, it will take sqrt2 = 1.414 times as long to travel the same distance.

To solve this problem, we need to understand the concept of vector addition and how it relates to the motion of the space probe.

When the space probe is moving in a straight line and both engines are firing in the same direction, the forces from the two engines act together to propel the spacecraft forward. The resulting force is the vector sum of the individual forces from each engine.

In this case, the probe takes 48 seconds to travel a certain distance. Since it starts from rest, we can assume that the acceleration of the probe is constant throughout this period.

Now, when the engines are fired simultaneously, but the forces they apply are perpendicular to each other, the situation is a bit different. We can use the concept of vector components to break down each force into its horizontal and vertical components.

Let's say the force generated by each engine is F and the angle between the forces is 90 degrees, making them perpendicular to each other.

In this case, the horizontal component of the force from each engine does not have any effect on the horizontal motion of the probe because the forces cancel each other out in that direction. The vertical components from each engine simply add up to the total force in the upward direction.

Now, since the probe starts from rest, its initial velocity is zero. The vertical force acting on the probe causes it to accelerate upwards. The speed of the probe in the vertical direction increases with time.

To determine the time taken to travel the same distance when the forces are perpendicular, we need to consider the vertical motion of the probe. We can use the basic equations of motion to solve for the time.

The equation for displacement (d) of an object under constant acceleration is given by:
d = ut + (1/2)at^2

Here, u is the initial velocity, a is the acceleration, and t is the time.

Since the probe starts from rest, its initial velocity (u) is zero. We can rearrange the equation to solve for time (t):

t = sqrt((2d) / a)

In this case, the vertical component of the force from the engines provides the acceleration for the probe. The total vertical force can be calculated using vector addition, which is the sum of the vertical components from the two engine forces.

Since the individual forces from each engine are equal, the total vertical force is simply twice the vertical component of one of the engine forces.

Using trigonometry, we know that the vertical component of the force from one engine is F * sin(90 degrees) = F.

Therefore, the total vertical force is 2F.

Now we can calculate the time it takes for the probe to travel the same distance when the forces are perpendicular:

t = sqrt((2d) / (2F))

Notice that the factor of 2 in the denominator cancel out, resulting in:

t = sqrt(d / F)

So, to find the time it takes to travel the same distance when the engines are fired simultaneously and the forces are perpendicular, you need to know the value of d (the distance traveled) and F (the force generated by each engine).

Simply substitute these values into the equation t = sqrt(d / F) to find the time.