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From the velocity-time graph you provided:

**(a) Direction of the y-axis:**
The y-axis is directed upwards. This conclusion comes from the fact that the initial velocity of the ball is negative, which indicates it's moving downwards at the beginning (since gravity pulls the ball down after being released). After hitting the ground, the velocity becomes positive, which means it is moving upwards.

**(b) Estimating the initial and final heights:**
To estimate the heights, we can use the kinematic equation that relates velocity and displacement under constant acceleration (gravity):
\[ v^2 = v_0^2 + 2a(y - y_0) \]

Here, \( v_0 \) and \( v \) are the initial and final velocities, \( y_0 \) and \( y \) are the initial and final positions, and \( a \) is the acceleration due to gravity, approximately \(-9.81 \, m/s^2\) (we use negative because gravity acts downwards).

1. **Initial height estimation:**
   - Initial velocity, \( v_0 \), can be estimated from the graph where the ball starts (approx. -4 m/s).
   - The ball is dropped (so \( v \) at the highest point is 0 m/s just before hitting the ground).
   - Solve for \( y_0 \) (initial height):
     \[
     0^2 = (-4)^2 + 2(-9.81)(y - 0)
     \]
     \[
     y = \frac{(-4)^2}{2 \times 9.81}
     \]

2. **Final height estimation:**
   - Final velocity just before reaching the final height is 0 m/s again.
   - The maximum upward velocity after the bounce is about 4 m/s (from the graph).
   - Solve for \( y \) (final height):
     \[
     0^2 = 4^2 + 2(-9.81)(y - 0)
     \]
     \[
     y = \frac{4^2}{2 \times 9.81}
     \]

Let's calculate these heights using the formula above.

Oliver drops a tennis ball from a certain height above a concrete floor. The ball's motion from the moment the ball is released to the moment it reaches its maximum height after bouncing up from the floor is shown on the velocity-time graph below. (a) Is the y-axis directed up or down? How do you know? (b) Estimate the initial height from which the ball is released and the final height to which the ball bounces.

Solve a kinematics problem where a tennis ball is dropped and bounces on a concrete floor. Use velocity-time graphs and kinematic equations to estimate initial and final heights based on the ball's motion. Perfect for high school physics students.

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