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This graph shows the uniformly accelerated motion of a ball in free fall on another planet, with time \( t \) on the x-axis (in seconds) and height \( h \) of the ball on the y-axis (in meters). The graph displays a parabolic shape, which suggests that the ball's height decreases as it falls under the influence of gravity, following the equation for motion under constant acceleration: 

\[
h(t) = h_0 - \frac{1}{2} g t^2
\]

where \( h_0 \) is the initial height, \( g \) is the acceleration due to gravity, and \( t \) is time.

### Steps to find acceleration due to gravity (\( g \)):
1. The graph shows that the initial height of the ball is about 4.9 m when \( t = 0 \).
2. The ball reaches the ground (height \( h = 0 \)) at around \( t = 0.5 \) seconds.
3. Using the kinematic equation for uniformly accelerated motion: 

\[
h(t) = h_0 - \frac{1}{2} g t^2
\]

Set \( h(t) = 0 \) when the ball reaches the ground, and substitute \( h_0 = 4.9 \, \text{m} \) and \( t = 0.5 \, \text{s} \):

\[
0 = 4.9 - \frac{1}{2} g (0.5)^2
\]

Now solve for \( g \):

\[
4.9 = \frac{1}{2} g \times 0.25
\]
\[
4.9 = 0.125 g
\]
\[
g = \frac{4.9}{0.125} = 39.2 \, \text{m/s}^2
\]

Thus, the acceleration due to gravity on this planet is approximately \( 39.2 \, \text{m/s}^2 \).

Would you like further details on any part of this explanation?

Here are some related questions:
1. How is free fall motion described mathematically?
2. How would the graph look if the acceleration due to gravity were lower?
3. What other kinematic equations can describe free-fall motion?
4. How does gravity vary on different planets?
5. What experiments can be done to measure gravity in a similar setup?

**Tip:** In free-fall problems, the initial velocity is typically zero when an object is dropped from rest.

On another planet, a ball is in free fall after being released from rest at time t = 0. A graph of the height of the ball above the planet’s surface as a function of time t is shown. The acceleration due to gravity on the planet is most nearly ____?

In this problem, a ball is in free fall on another planet. Using the kinematic equation of motion and the graph provided, calculate the acceleration due to gravity. The ball's initial height is 4.9 meters, and it reaches the ground in 0.5 seconds, giving an approximate gravity of 39.2 m/s².

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