A ball is shot vertically upward from the surface of another planet. A plot of y versus t for the ball is shown in the figure below, where y is the height of the ball above its starting point and t = 0 at the instance the ball is shot. The figure’s vertical scale is set by ys = 30.0 m. (a) Write expressions for the displacement and velocity of the ball as functions of time. (b) What is the magnitude of the free-fall acceleration on the planet? (c) What is the magnitude of the initial velocity of the ball?

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To answer these questions, we need to analyze the information given in the figure. Let's break down the questions one by one.

(a) Write expressions for the displacement and velocity of the ball as functions of time:
To determine the expressions for displacement and velocity, we need to examine the characteristics of the plot. From the provided figure, we can see that the ball starts at y = 0 and reaches a maximum height of y = +30.0 m before returning to its starting point. This indicates that the ball undergoes free fall motion.

The displacement of the ball as a function of time can be written as follows:
For the ascending phase (upwards motion):
y = y₀ + v₀t - (1/2)gt²

For the descending phase (downwards motion):
y = y₀ + v₀t + (1/2)gt²

Here, y₀ is the initial height of the ball (zero in this case), v₀ is the initial velocity of the ball, t is the time, and g is the acceleration due to gravity on the planet.

The velocity of the ball as a function of time can be obtained by differentiating the displacement equation with respect to time:

For the ascending phase:
v = v₀ - gt

For the descending phase:
v = v₀ + gt

(b) What is the magnitude of the free-fall acceleration on the planet?
The magnitude of the free-fall acceleration is given by the acceleration due to gravity. From the plot, we can see that the ball takes the same time to reach its peak and fall back to its starting point. Therefore, we can calculate the time it took to reach the peak and use it to determine the acceleration.

(c) What is the magnitude of the initial velocity of the ball?
To find the magnitude of the initial velocity, we need to determine the velocity at the beginning of the motion. This can be obtained from the velocity equation above by substituting t = 0.

To find the exact values of the time, acceleration, and initial velocity, we would need additional information from the figure or problem statement.

(a) To write expressions for the displacement and velocity of the ball as functions of time, we need to analyze the graph provided.

We can see from the graph that the ball initially moves upward and then reaches its highest point before falling back down. Therefore, we can divide the motion into two parts: the upward motion and the downward motion.

During the upward motion:
- The displacement (y) of the ball can be represented as the vertical distance traveled by the ball from its starting point. As the ball moves upward, y increases. Therefore, the displacement can be given by y = f(t), where f(t) is a function of time.
- The velocity (v) of the ball can be calculated as the rate of change of displacement with respect to time. So, v = dy/dt, where dy is the change in displacement and dt is the change in time.

During the downward motion:
- The displacement (y) of the ball can be represented as the vertical distance traveled by the ball from its highest point to its starting point. As the ball moves downward, y decreases. Therefore, the displacement can be given by y = -g(t-t_peak) + h_peak, where g is the magnitude of the free-fall acceleration, t_peak represents the time when the ball reaches its highest point, and h_peak is the height of the ball at its highest point.
- The velocity (v) of the ball can be calculated as the rate of change of displacement with respect to time. So, v = dy/dt, where dy is the change in displacement and dt is the change in time.

(b) To find the magnitude of the free-fall acceleration on the planet, we need to determine the slope of the line representing the downward motion.

From the graph, the slope of the straight line during the downward motion represents the velocity of the ball. Since the velocity is constant during free-fall, the slope remains the same. Thus, we can find the magnitude of the free-fall acceleration by calculating the slope of the downward motion line.

(c) To find the magnitude of the initial velocity of the ball, we need to determine the slope of the line representing the upward motion.

From the graph, the slope of the straight line during the upward motion represents the velocity of the ball. Since the velocity is constant during upward motion, the slope remains the same. Thus, we can find the magnitude of the initial velocity by calculating the slope of the upward motion line.