A box is dropped from a spacecraft moving horizontally at 27.0 m/s at a distance of 155 m above the surface of a moon. The rate of freefall acceleration on this airless moon is 2.79 m/s2. (a) How long does it take for the box to reach the moon's surface? (b) What is its horizontal displacement during this time? (c) What is its vertical velocity when it strikes the surface? (d) At what speed does the box strike the moon?

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To solve this problem, we can use the equations of motion. Let's go step by step.

(a) To find the time it takes for the box to reach the moon's surface, we can use the equation:

\[ s = ut + \frac{1}{2}at^2 \]

where:
s = displacement (in this case, the distance from the spacecraft to the moon's surface)
u = initial velocity (0 m/s since the box is dropped)
a = acceleration (rate of freefall acceleration on the moon)
t = time

Rearranging the equation, we have:

\[ t = \sqrt{\frac{2s}{a}} \]

Substituting the given values, we have:

\[ t = \sqrt{\frac{2 \times 155}{2.79}} \]

Calculating, we find:

\[ t \approx 7.87 \, \text{s} \]

So it takes approximately 7.87 seconds for the box to reach the moon's surface.

(b) To find the horizontal displacement during this time, we can use the equation:

\[ d = vt \]

where:
d = displacement (horizontal distance traveled)
v = horizontal velocity (27.0 m/s)
t = time (7.87 s)

Substituting the given values, we have:

\[ d = 27.0 \times 7.87 \]

Calculating, we find:

\[ d \approx 212.49 \, \text{m} \]

So its horizontal displacement during this time is approximately 212.49 meters.

(c) To find the vertical velocity when it strikes the surface, we can use the equation:

\[ v = u + at \]

where:
v = final velocity
u = initial velocity (0 m/s since the box is dropped)
a = acceleration (rate of freefall acceleration on the moon)
t = time (7.87 s)

Substituting the given values, we have:

\[ v = 0 + 2.79 \times 7.87 \]

Calculating, we find:

\[ v \approx 21.95 \, \text{m/s} \]

So its vertical velocity when it strikes the surface is approximately 21.95 m/s.

(d) To find the speed at which the box strikes the moon, we can use the Pythagorean theorem:

\[ \text{Speed} = \sqrt{\text{Vertical Velocity}^2 + \text{Horizontal Velocity}^2} \]

where:
Vertical Velocity = 21.95 m/s (from part c)
Horizontal Velocity = 27.0 m/s

Substituting the given values, we have:

\[ \text{Speed} = \sqrt{21.95^2 + 27.0^2} \]

Calculating, we find:

\[ \text{Speed} \approx 34.92 \, \text{m/s} \]

So the box strikes the moon at a speed of approximately 34.92 m/s.

To answer these questions, we can use the kinematic equations. Here's how you can find the answers:

(a) How long does it take for the box to reach the moon's surface?

We can use the equation:
displacement = initial velocity * time + (1/2) * acceleration * time^2

In this case, the initial velocity is 0 m/s (since the box is dropped), the displacement is 155 m, and the acceleration is 2.79 m/s^2 (freefall acceleration on the moon). Setting up the equation and solving for time will give us the answer.

155 = 0*t + (1/2)*2.79*t^2

Rearranging and solving for t:

2.79*t^2 = 155*2

t^2 = (155*2) / 2.79

t = sqrt((155*2) / 2.79)

Calculating this value will give us the time it takes for the box to reach the moon's surface.

(b) What is its horizontal displacement during this time?

The horizontal velocity of the box remains constant at 27.0 m/s. Therefore, the horizontal displacement is given by:

horizontal displacement = horizontal velocity * time

Simply multiply the horizontal velocity by the time calculated in question (a) to find the answer.

(c) What is its vertical velocity when it strikes the surface?

The vertical velocity can be calculated using the equation:
final velocity = initial velocity + acceleration * time

The initial velocity is 0 m/s (since the box is dropped), the acceleration is 2.79 m/s^2, and the time is the one calculated in question (a). Plug these values into the equation and solve for the final velocity.

(d) At what speed does the box strike the moon?

The speed is simply the magnitude of the final velocity. So, take the absolute value of the final velocity calculated in question (c) to get the answer.

By following these steps, you should be able to find the answers to all the questions.