The acceleration of gravity at the surface of a newly discovered planet is 5.2 m/s^2. An astronaut throws a rock straight up at 10.5 m/s on this planet. How high will it go?

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To find the height the rock will reach, we need to consider the principles of motion and the equations of motion. The equation that relates the final position (height) reached by an object thrown upward, its initial velocity, and the acceleration due to gravity is:

hf = hi + vi * t - 0.5 * g * t^2

Where:
- hf is the final height (what we want to find),
- hi is the initial height (which we assume to be zero in this case, as the rock is thrown from the surface),
- vi is the initial velocity of the rock (10.5 m/s),
- t is the time it takes for the rock to reach its highest point (which we will solve for),
- g is the acceleration due to gravity (5.2 m/s^2).

In this case, since the rock is thrown straight up, it will reach its highest point when its vertical velocity becomes zero. We can use this fact to find the time it takes for the rock to reach its highest point.

The vertical velocity of the rock, v(t), can be determined using the equation:

v(t) = vi - g * t

When the rock reaches its highest point, v(t) will be zero. Therefore, we can solve for t:

0 = vi - g * t

Rearranging the equation:

t = vi / g

Now we can substitute the values:

t = 10.5 m/s / 5.2 m/s^2

Simplifying:

t ≈ 2.019 seconds

Now we can substitute the value of t back into the first equation to find the final height:

hf = hi + vi * t - 0.5 * g * t^2
= 0 + (10.5 m/s) * (2.019 s) - 0.5 * (5.2 m/s^2) * (2.019 s)^2

Calculating:

hf ≈ 10.54694 meters

Therefore, the rock will reach a height of approximately 10.55 meters on this newly discovered planet.