An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 10.0 m if her initial speed is 2.60 m/s. What is the free-fall acceleration on the planet?
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To solve this problem, we can use the kinematic equation for horizontal distance:
d = v₀t + (1/2)at²
where:
d = horizontal distance
v₀ = initial speed
t = time
a = acceleration
In this case, the astronaut can jump a maximum horizontal distance of 10.0 m, and her initial speed is 2.60 m/s. We want to find the free-fall acceleration (a) on the planet.
In order to use the above equation, we need to find the time it takes for the astronaut to cover the maximum horizontal distance.
Using the equation of motion for horizontal displacement:
d = v₀t
Substituting the given values:
10.0 m = 2.60 m/s * t
Solving for t, we get:
t = 10.0 m / 2.60 m/s
t ≈ 3.85 s
Now, we can substitute the value of t into the first equation to find the acceleration (a):
10.0 m = (2.60 m/s)(3.85 s) + (1/2)a(3.85 s)²
Simplifying the equation:
10.0 m = 10.01 m + (1.92 s²)a
Rearranging the equation:
10.0 m - 10.01 m = (1.92 s²)a
-0.01 m = (1.92 s²)a
Solving for a:
a = -0.01 m / (1.92 s²)
a ≈ -0.0052 m/s²
Therefore, the free-fall acceleration on the planet is approximately -0.0052 m/s². The negative sign indicates that the astronaut is experiencing a deceleration in the horizontal direction on the planet.
To find the free-fall acceleration on the planet, we can use the kinematic equation for horizontal motion:
d = v_initial * t + (1/2) * a * t^2
In this case, the maximum horizontal distance (d) is given as 10.0 m, the initial speed (v_initial) is 2.60 m/s, and we want to find the acceleration (a).
Since the astronaut is jumping horizontally, we know the time (t) it takes for her to reach the maximum distance will be the same as the time it takes for her to land. Therefore, we can ignore the first term (v_initial * t) in the equation.
Now, we can rearrange the equation to solve for the acceleration:
d = (1/2) * a * t^2
2d = a * t^2
a = (2d) / t^2
To solve for the acceleration, we still need to find the time it takes for the astronaut to jump horizontally. We can calculate this using the equation for horizontal motion:
d = v * t
10.0 m = 2.60 m/s * t
t = 10.0 m / 2.60 m/s
t ≈ 3.846 s
Now, we can substitute the values of d and t into the acceleration equation to get the answer:
a = (2 * 10.0 m) / (3.846 s)^2
a ≈ 1.086 m/s^2
Therefore, the free-fall acceleration on the strange planet is approximately 1.086 m/s^2.