91Ó°ÊÓ

When discussing the motion of objects like a baseball being hit straight up, the initial velocity is a crucial factor. It determines how quickly an object is launched from the start.
In kinematics, the initial velocity often plays a significant role in predicting future motion. For the baseball problem, we are interested in finding how quickly the ball was moving right after being hit by the bat.
To find the initial velocity, we utilize the kinematic equation:

• \(v_f = v_i + at\)

- Here, \(v_f\) is the final velocity (0 m/s at the maximum height), \(v_i\) is what we're solving for, \(a\) is the acceleration due to gravity (\(-9.8 \,\mathrm{m/s^2}\)), and \(t\) is the time (3 seconds).
- Plugging in these values, we rearrange the equation to solve for \(v_i\): \[v_i = v_f - at = 0 - (-9.8 \,\mathrm{m/s^2}) \times 3.0 \,\mathrm{s} = 29.4 \,\mathrm{m/s}\\] Hence, the initial velocity of the baseball is \(29.4 \,\mathrm{m/s}\).

Achieving the maximum height is the point when the baseball stops moving upward and is about to start its descent. At this peak, its velocity becomes zero. Calculating this height gives insight into how "high" the ball actually travels.
To find the maximum height, we apply another important equation from kinematics:

• \(h = v_i t + 0.5 a t^2\)

- We use the previously calculated initial velocity \(29.4 \,\mathrm{m/s}\), time \(3.0 \,\mathrm{s}\), and known acceleration \(-9.8 \,\mathrm{m/s^2}\).
- Substituting these values, the equation becomes:\[h = (29.4 \,\mathrm{m/s}) \times 3.0 \,\mathrm{s} + 0.5 \times (-9.8 \,\mathrm{m/s^2}) \times (3.0 \,\mathrm{s})^2\\]After performing the arithmetic, we find that the maximum height \(h\) is \(44.1 \,\mathrm{m}\). Therefore, the baseball reaches a height of \(44.1 \,\mathrm{m}\) before reversing its motion.

The acceleration due to gravity is a constant that affects all objects moving under the Earth's gravitational field. It is a downward force that pulls objects towards the center of the Earth, measured as \(9.8 \,\mathrm{m/s^2}\).
In the context of our problem, this value is crucial as it dictates the motion of the baseball after being hit.

• The negative sign in \(-9.8 \,\mathrm{m/s^2}\) indicates that gravity acts in the opposite direction to the initial upward motion.

Understanding gravity's role helps explain why objects slow down as they reach their highest point and eventually start to fall. Thus, it is this constant acceleration that influences the ball's upward motion, causing it to decelerate until it reaches the maximum height where its velocity is zero. This understanding allows us to predict the ball's trajectory and height effectively.