Question

Consider the flight of a tennis ball with mass \(57 \mathrm{~g}\) and a diameter of \(66.0 \mathrm{~mm}\). Assume the ball is served \(6.40\) meters from the net at a speed of \(50.0 \mathrm{~m} / \mathrm{s}\) down the center line from a height of \(2.8 \mathrm{~m}\). It needs to just clear the net \((0.914 \mathrm{~m})\). a. Ignoring air resistance and spin, analytically find the path of the ball assuming it just clears the net. Determine the angle to clear the net and the time of flight. b. Find the angle to clear the net assuming the tennis ball is given a topspin with \(w=50 \mathrm{rad} / \mathrm{s}\) c. Repeat part b assuming the tennis ball is given a bottom spin with \(\omega=50 \mathrm{rad} / \mathrm{s}\) d. Repeat parts \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\) with a drag force, taking \(C_{D}=0.55\).

Short answer

The computations involved in this exercise are quite complex and depend on the specific numbers for the given variables. The analytical solution would involve applying the equations of motion, the Magnus effect and the equation for drag force, and solving for the trajectory, angle and time of flight of the tennis ball under each different condition.

Step-by-step solution

Calculating path of the ball ignoring air resistance and spin

First we will find the path of the ball just ignoring the air resistance and spin. We will use the basic equations of motion. From this, the time of flight \(t\) and angle \(\theta\) can be calculated using the equations: \(t = \frac{y_{i} - y_{f}}{v_{i}sin(\theta)}\) and \(\theta = arcsin(\frac{d_{f}-d_{i}}{v_{i}t})\) respectively where \(y_{i}\) and \(y_{f}\) are initial and final heights, \(d_{i}\) and \(d_{f}\) are initial and final distances.

Calculating angle with top spin

Now, we introduce top spin. The physics behind a top spin is the Magnus effect, which will cause the ball to dip sooner than it would with no spin. The force produced by the spin will alter the trajectory of the ball and will cause it to travel in a curve, rather than a straight line. We can use the magnus force equation to find the new path and then recalculate the angle using the similar method from step 1.

Calculating angle with bottom spin

Next, we introduce bottom spin. Similarly to top spin, bottom spin will impact the flight path of the ball due to the Magnus effect. The effect of bottom spin is just the opposite of top spin--the ball will tend to stay in the air longer than with no spin. Again, use the Magnus force equation for the new force and recalculate the angle.

Introducing drag force and repeating previous parts

Lastly, we incorporate drag force. Drag force is a resistive force that acts opposite to the direction of motion. The effect of air resistance will reduce the horizontal distance traveled by the ball, and slightly affect the path as well. For all parts a, b, and c, we add drag force \( F_{drag} = 0.5 * \rho * C_{D} * A * v^2 \) where \(\rho\) is the air density, \(C_{D}\) is the drag coefficient and \(A\) is the surface area, and repeat the calculations of the path, angle and the time of flight.

Key concepts

Projectile Motion

Understanding how tennis balls move through the air on the court involves a concept called projectile motion. It's a type of motion experienced by an object that is thrown near the Earth's surface and moves along a curved path under the action of gravity only.

In tennis, when a player serves the ball, the ball moves through the air in a parabolic trajectory before it lands on the opposing side. The factors that determine this path are the ball's initial velocity, the angle of launch, and the acceleration due to gravity. The initial speed and angle of the serve combine to give the ball a certain amount of kinetic energy, which is converted into gravitational potential energy as the ball rises, then back to kinetic as it falls.

If we ignore air resistance, the horizontal component of the ball's velocity will remain constant, and the vertical component will be affected solely by gravity. The mathematical formulae to calculate the time of flight and the angle to clear the net, without considering the air resistance and spin, are crucial for players to know how to serve the ball successfully over the net.

Magnus Effect

Ever noticed how a tennis ball can sometimes curve through the air, or dip and kick unexpectedly? That's the Magnus effect in action. It's named after Heinrich Gustav Magnus, the physicist who investigated this phenomenon.

The Magnus effect comes into play when a ball is spinning as it moves through the air. As it spins, it pushes air in one direction, and, in reaction, a force is exerted in the opposite direction. For a topspin serve, the top of the ball moves with the direction of the flight, pushing air down and creating an upward lift force on the bottom of the ball. This makes the ball dip down sooner than it would without spin. Conversely, a ball with backspin (or bottom spin) will create lift force opposite to the direction of gravity, allowing the ball to stay in the air longer.

Tennis players use the Magnus effect to their advantage by putting different spins on the ball to control its trajectory and make it more difficult for their opponents to predict where the ball will land.

Drag Force

Imagine trying to move your hand through water; you can feel the resistance it offers. That's similar to the drag force acting on a tennis ball as it cuts through the air. Drag force works against the motion of the ball and is a type of friction that reduces its speed.

In tennis, drag force is pivotal because it affects how far and fast the ball will travel. The drag depends on several factors, including the speed of the ball, its cross-sectional area, the density of the air, and the drag coefficient, which is a number that represents how aerodynamic the object is. The equation for drag force is given by\( F_{drag} = 0.5 * \rho * C_{D} * A * v^2 \), where \( \rho \) is the air density, \( C_{D} \) is the drag coefficient, and \( A \) is the surface area of the ball.

Understanding the drag force is important for players when it comes to adjusting their serves and shots to compensate for the slower speeds, especially on high-speed serves or when playing in windy conditions.