An astronaut on a distant planet wants to determine its acceleration due to gravity. The astronaut throws a rock straight up with a velocity of + 14.2 m/s and measures a time of 22.2 s before the rock returns to his hand. What is the acceleration (magnitude and direction) due to gravity on this planet? (positive = up, negative = down)
Well, it seems like this astronaut really wants to rock and roll on this distant planet! Let's help them out with some calculations.
To determine the acceleration due to gravity, we can use the equation:
vf = vi + at
Here, vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time.
Since the rock is thrown straight up, we know that the final velocity when it returns to the astronaut's hand is 0 m/s. The initial velocity is +14.2 m/s (positive because it's moving up). The time taken is 22.2 seconds.
So, we can rewrite the equation as:
0 = 14.2 + a x 22.2
Now let's solve this equation for a, the acceleration:
a x 22.2 = -14.2
a = -14.2 / 22.2
a ≈ -0.64 m/s²
So, the magnitude of the acceleration due to gravity on this planet is approximately 0.64 m/s², and since it's negative, it means the direction is down. Looks like this astronaut will have to watch out for falling space bananas!
To find the acceleration due to gravity on the distant planet, we can use the following steps:
Step 1: Identify the known variables:
- Initial velocity (u) = +14.2 m/s (upwards)
- Time taken (t) = 22.2 s
Step 2: Determine the displacement during the upward and downward motion of the rock:
Since the rock is thrown straight up and returns to the astronaut's hand, the displacements during upward and downward motion are equal in magnitude but opposite in direction. Therefore, we can consider only the upward displacement.
Displacement (s) = ?
Step 3: Use the equation for displacement:
The equation for displacement can be given by:
s = u*t + (1/2)*a*t^2
Since we are only considering the upward motion, the acceleration due to gravity is negative (opposite direction to the initial velocity).
Step 4: Substitute the known values into the equation:
s = (+14.2 m/s) * (22.2 s) + (1/2) * (a) * (22.2 s)^2
Step 5: Solve for the displacement:
We need to calculate 's' to continue with the next steps.
Step 6: Simplify the equation:
s = 314.04 m + 11.1 a
Step 7: Determine the time of flight for the entire motion:
The total time of flight (T) can be given by:
T = t + t = 2t = 2 * 22.2 s = 44.4 s
Step 8: Use the equation for displacement during free fall:
The equation for displacement during free fall can be given by:
s = (1/2) * a * T^2
Step 9: Substitute the known values into the equation:
s = (1/2) * (a) * (44.4 s)^2
Step 10: Simplify the equation:
s = 0.5 * 1959.84 a
Step 11: Equate the two expressions for s:
314.04 m + 11.1a = 0.5 * 1959.84 a
Step 12: Solve for 'a':
Rearrange the equation to solve for 'a':
314.04 m = 0.5 * 1959.84 a - 11.1a
314.04 m = 979.92 a - 11.1a
314.04 m = 968.82 a
a = (314.04 m) / (968.82)
Step 13: Calculate 'a':
Using a calculator or performing the calculation, we find:
a ≈ 0.3234 m/s^2
Step 14: Determine the direction of acceleration:
Since the acceleration is opposite to the initial velocity, the direction of acceleration due to gravity on this planet would be downward or negative.
Therefore, the acceleration (magnitude and direction) due to gravity on this planet is approximately 0.3234 m/s^2 downward.
To determine the acceleration due to gravity on a distant planet, we can use the equation of motion for an object in free fall:
y = v₀t + (1/2)at²
where:
y = displacement (change in height),
v₀ = initial velocity,
t = time,
a = acceleration due to gravity.
In this case, the astronaut throws the rock straight up, so the initial velocity (v₀) is +14.2 m/s (positive, because it is directed upwards). The time it takes for the rock to return to the astronaut's hand, t, is 22.2 s.
When the rock returns to the astronaut's hand, its displacement is zero (y = 0) since it is at the same height. We can plug in the values into the equation:
0 = (+14.2 m/s)(22.2 s) + (1/2)a(22.2 s)²
Now, we can solve for the acceleration due to gravity (a):
-14.2 m/s * 22.2 s = (1/2)a(22.2 s)²
-314.04 m²/s² = (1/2)a (491.84 s²)
Now, we can isolate "a" by dividing both sides of the equation by (1/2)(491.84 s²):
-314.04 m²/s² / (1/2)(491.84 s²) = a
a ≈ -1.28 m/s² (rounded to two decimal places)
The magnitude of the acceleration due to gravity on this planet is approximately 1.28 m/s², and since the negative sign indicates the direction (negative = down), the direction of gravity on this planet is downwards.