Problem 53 - College Physics

Question

Human biomechanics. The fastest served tennis ball, served by "Big Bill" Tilden in \(1931,\) was measured at 73.14 \(\mathrm{m} / \mathrm{s}\) . The mass of a tennis ball is \(57 \mathrm{g},\) and the ball is typically in contact with the tennis racquet for 30.0 \(\mathrm{ms}\) , with the ball starting from rest. Assuming constant acceleration, (a) what force did Big Bill's tennis racquet exert on the tennis ball if he hit it essentially horizontally? (b) Make free- body diagrams of the tennis ball during the serve and just after it has moved free of the racquet.

Step-by-Step Solution

Answer

The force exerted by the racquet is 139.0 N. Diagrams A and B depict forces during and after contact.

Step 1: Convert Mass and Time Units

The mass of the tennis ball is given in grams. We need to convert it to kilograms.\[ 57 \text{ g} = 57 \times 10^{-3} \text{ kg} = 0.057 \text{ kg}.\]The contact time is given in milliseconds. We need to convert it to seconds.\[ 30.0 \text{ ms} = 30.0 \times 10^{-3} \text{ s} = 0.03 \text{ s}. \]

Step 2: Calculate the Acceleration

The initial velocity \(u\) is 0 m/s (since the ball starts from rest), and the final velocity \(v\) is 73.14 m/s. We need to calculate the acceleration \(a\) using the formula:\[ a = \frac{v - u}{t} = \frac{73.14 \text{ m/s} - 0}{0.03 \text{ s}} = 2438 \text{ m/s}^2. \]

Step 3: Calculate the Force Exerted by the Racquet

With the mass \(m\) in kg and the acceleration \(a\) in m/s² known, we use Newton's second law \(F = ma\) to find the force \(F\) exerted by the racquet.\[ F = m \times a = 0.057 \text{ kg} \times 2438 \text{ m/s}^2 = 139.0 \text{ N}. \]

Step 4: Draw Free-Body Diagrams

**Diagram A** (while in contact with the racquet): - **Forces acting on the ball:** - Horizontal force due to the racquet. - Negligible air resistance due to the short contact time. **Diagram B** (just after losing contact with the racquet): - **Forces acting on the ball:** - Gravity acting downward (9.8 m/s²). - Negligible air resistance before the ball significantly slows down.

Key Concepts

Tennis Physics

When it comes to tennis, you might not immediately think of physics. However, tennis is a fantastic example of physics in action! Every time a player hits the ball, they are essentially applying principles of mechanics to it. The speed, spin, and direction of the tennis ball are influenced by these physical laws.
In the case of "Big Bill" Tilden's serve, both speed and force play crucial roles. The serve not only needs to be fast to surprise opponents, but it should also be accurate and powerful to win points. Understanding physics helps players fine-tune their serves and shots for better performance.
Plus, physics explains why tennis balls behave the way they do during other plays, like bouncing or spinning. Physics isn't just a lesson from a book; it makes the game of tennis possible!
Studying the motion of a tennis ball involves examining forces, speeds, and how they interact to influence the ball’s trajectory and velocity.

Newton's Second Law

Newton's Second Law is pivotal in understanding how forces affect the movement of objects, especially in sports like tennis. This law states that the force acting on an object is equal to the mass of that object times its acceleration ( F = ma ).
In Bill Tilden's serve, this law helps calculate the force exerted on the tennis ball by the racquet. When the tennis ball, initially at rest, is struck, it accelerates to its final velocity. The mass of the ball and the acceleration experienced during contact determines the force generated.
- **Force ( F ):** This is what you need to push or pull the tennis ball. - **Mass ( m ):** The amount of matter in the tennis ball, here it's 0.057 kg. - **Acceleration ( a ):** The change in velocity that the ball experiences, which is 2438 m/s² in this situation.
Understanding this law allows players to comprehend how varying the force or acceleration factors can affect gameplay. It shows the relationship between these elements and how they contribute to the ball's movement.

Free-Body Diagrams

Free-body diagrams are incredibly useful in visualizing the forces at play. They break down the complex interactions into simple diagrams, which makes it easier to understand the physics involved.
In any given situation, a free-body diagram isolates a single object and represents all the forces acting on it:

• **During Contact:** When the ball is in contact with the racquet, the primary force is the push from the racquet. Though theoretically, air resistance exists, it's negligible due to the short interaction time.
• **Post Contact:** Once the ball leaves the racquet, gravity is the main force acting on it, pulling the ball towards the ground. Air resistance isn't relevant until the ball's velocity decreases significantly over time and distance.

By sketching these diagrams, you gain a clearer understanding of how each force influences the ball's motion during and after the hit. This knowledge forms the basis for more complex analyses in mechanics.

Constant Acceleration

Constant acceleration refers to situations where the acceleration doesn't change over the time period in question. In the context of the tennis serve problem, we assume the acceleration remains constant while the racquet is in contact with the tennis ball.
This assumption simplifies calculations, as it allows us to use straightforward equations of motion. The formula used is:\[a = \frac{v - u}{t}\]- **\(v\):** Final velocity (73.14 m/s).- **\(u\):** Initial velocity (0 m/s, the ball starts from rest).- **\(t\):** Time of contact (0.03 s).
Knowing that acceleration is constant helps predict how fast the ball will be moving at any given time during the serve. Without this concept, our understanding of how the ball speeds up during the serve would be much less precise.

Force Calculation

Calculating force is a crucial component in understanding any physical activity. In "Big Bill" Tilden's historical serve, determining the force exerted by the racquet offers insights into the mechanics behind powerful serves.
Using the given acceleration and the mass of the ball, Newton's Second Law ( F = ma ) allows us to compute the force. In this scenario:

• Mass ( m ) = 0.057 kg
• Acceleration ( a ) = 2438 m/s²
• Force ( F ) = m × a = 139.0 N

Thus, 139 Newtons is the force necessary for the racquet to accelerate the tennis ball to its top speed.
Reading into force calculation helps players and coaches strategize on using optimal strengths and techniques to enhance serves and swings during the game, which can dramatically affect performance.