91Ó°ÊÓ

In kinematics, constant acceleration means that an object's velocity changes at a steady rate over time. When acceleration is constant, it simplifies the calculations since the variables involved change linearly. Instead of acceleration fluctuating with forces, as they often do in real life, constant acceleration assumes a uniform force is applied.
This concept is central to many physics problems because expressions for velocity, distance, and time can be easily derived. For the tennis ball serve, because the acceleration is steady once the racket contacts the ball, it enables us to calculate how quickly the ball reaches its final velocity and the distance it covers during that time.

Initial velocity is the speed at which an object begins its motion, in this case, the tennis ball. It can either be zero or any positive number. Here, we’re told the tennis ball starts from rest, indicating its initial velocity is 0 m/s.
This information is crucial, as the starting speed impacts how we compute subsequent motion properties. Knowing the initial velocity allows us to calculate the acceleration accurately using the kinematic equations. For the tennis serve problem, since the ball begins from a standstill, the calculations for reach speed and distance are straightforward, focusing primarily on acceleration and final velocity.

The distance traveled by an object under constant acceleration can be found using the equation:

• \( d = v_i t + \frac{1}{2} a t^2 \)

The formula incorporates the initial velocity, time, and acceleration. Here, with the initial velocity being zero, the term simplifies since the entire movement is due to the acceleration.
In the served tennis ball scenario, we calculate the distance the ball moves while under the racquet's influence. By applying the known values into the equation, where \( a = 2438 \text{ m/s}^2 \) and \( t = 0.030 \) s, the ball travels 1.1 meters. This measurement helps understand how far the ball progresses during the brief contact time.

The final velocity is the speed an object attains at the end of a time duration. It’s crucial for understanding how fast the tennis ball travels after being struck by the racket.
When calculating the final velocity in scenarios assuming constant acceleration, we use the formula:

• \( v_f = v_i + a t \)

Even though the final velocity was given as part of the problem (73.14 m/s), knowing the initial velocity and time of impact enables one to verify this velocity.
Thus, using the formula ensures that calculated dynamics align with the observed results in reality, validating both given assumptions and derived conclusions.

The time of contact highlights the duration over which an object's velocity changes. In this particular case, the tennis ball remains in touch with the racket for 30 milliseconds, equivalent to 0.030 seconds.
This time span is a key factor in calculating both acceleration and distance. Since force is applied only during contact, it defines the total time for which the ball experiences acceleration.
Short contact times, like those in this example, necessitate rapid changes in velocity, leading to high accelerations. These brief intervals illustrate how effectively forces can alter motion, particularly in sports dynamics, where milliseconds define outcomes.