91Ó°ÊÓ

The concept of change in velocity is important when considering objects in motion. Velocity refers to the speed and direction of an object. When a baseball is pitched horizontally and then hit upwards, its velocity changes in magnitude and direction. This change is crucial for understanding how the force applied by the bat affects the baseball's motion.

In our original exercise, the baseball initially moves with a horizontal velocity of 27.0 m/s. When it is hit by the bat, it travels upwards with a velocity of 24.86 m/s. To find the change in velocity (\( \Delta v \)), you subtract the initial velocity from the final velocity.

- The initial horizontal velocity, \(v_i\), is -27.0 m/s (indicating direction directly opposite to the final direction).- The final vertical velocity, \(v_f\), is 24.86 m/s (upwards).
Adding these gives:

• \( \Delta v = 24.86 - (-27.0) = 51.86 \ \text{m/s} \)

This number represents how drastically the ball's speed and direction changed due to the impact.

Kinetic and potential energy are key concepts in physics, explaining energy forms associated with motion and position.

**Kinetic Energy (KE):** This is the energy due to motion. An object has more kinetic energy when it moves faster. It's calculated with the formula:\[KE = \frac{1}{2}mv^2\]

**Potential Energy (PE):** This is energy stored by an object's position. In the case of our baseball, it achieves potential energy when it reaches a height of 31.5 m after being hit.We use:\[PE = mgh\]where \(m\) is mass, \(g\) is gravity, and \(h\) is height.

In the exercise, when the baseball leaves the bat, its kinetic energy leads it to convert into potential energy as it rises, showing the conservation of energy principle:

• The potential energy at the maximum height equals the kinetic energy as it leaves the bat.
• \(PE = KE \rightarrow mgh = \frac{1}{2}mv^2\)
• Solving this gives the baseball's exit velocity, \(v \approx 24.86 \ \text{m/s}\).

Understanding average force involves applying the impulse-momentum theorem. When force acts upon an object over a time interval, it changes the object's momentum.

Momentum (\(p\)) represents mass in motion. Force alters an object's velocity, hence its momentum. The impulse-momentum theorem is expressed by:

• \(F \Delta t = m \Delta v\)

In this formula, \(F\) is force, \(\Delta t\) is the time duration, \(m\) is mass, and \(\Delta v\) is the change in velocity.

In the provided scenario, the baseball's mass is 0.145 kg, and the contact time with the bat is 2.5 ms (\(2.5 \times 10^{-3} \ \text{s}\)). With a calculated \(\Delta v\) of 51.86 m/s, we find the average force, \(F\), by rearranging the formula:

• \(F = \frac{m \Delta v}{\Delta t}\)
• \(F = \frac{0.145 \times 51.86}{2.5 \times 10^{-3}} \approx 3010.44 \ \text{N}\)

This result highlights the significant force the bat exerted in a very short time to reverse the baseball's direction and speed.