Question

You serve a tennis ball from a height of \(1.8 \mathrm{~m}\) above the ground. The ball leaves your racket with a speed of \(18.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(7.00^{\circ}\) above the horizontal. The horizontal distance from the court's baseline to the net is \(11.83 \mathrm{~m},\) and the net is \(1.07 \mathrm{~m}\) high. Neglect spin imparted on the ball as well as air resistance effects. Does the ball clear the net? If yes, by how much? If not, by how much did it miss?

Short answer

Answer: To determine if the tennis ball clears the net and by how much, follow the steps provided in the solution. Calculate the initial horizontal and vertical velocities, the time it takes to reach the net horizontally, and the vertical position when it reaches the net. Compare the calculated vertical position to the height of the net to find if it clears the net and by how much.

Step-by-step solution

Identify the given information

We are given the following information: - Initial height of the ball: \(1.8 \mathrm{~m}\) - Initial speed of the ball: \(18.0 \mathrm{~m} / \mathrm{s}\) - Launch angle: \(7.00^{\circ}\) - Distance to the net: \(11.83 \mathrm{~m}\) - Height of the net: \(1.07 \mathrm{~m}\)

Calculate the initial horizontal and vertical velocities

Using the launch angle and the initial speed, we can find the initial horizontal and vertical components of velocity using the trigonometric functions sine and cosine: - Initial horizontal velocity (\(v_{0x}\)) = \(18.0 \mathrm{~m/s} \times \cos (7.00^{\circ})\) - Initial vertical velocity (\(v_{0y}\)) = \(18.0 \mathrm{~m/s} \times \sin (7.00^{\circ})\)

Calculate the time it takes to reach the net horizontally

We can find the time it takes for the ball to travel horizontally to the net by dividing the distance to the net by the initial horizontal velocity. Let's represent this time as \(t\). \(t = \frac{11.83 \mathrm{~m}}{v_{0x}}\)

Calculate the vertical position when it reaches the net

Using the time, \(t\), we can find the vertical position, \(y\), when the ball reaches the net. For this, we'll use the equation \(y = y_0 + v_{0y}t - \frac{1}{2}gt^2\), where \(y_0\) is the initial height, \(v_{0y}\) is the initial vertical velocity, and \(g\) is the acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)). \(y = 1.8 \mathrm{~m} + v_{0y}t - \frac{1}{2}(9.81 \mathrm{~m/s^2})t^2\)

Compare the calculated vertical position to the height of the net

Now that we have the vertical position of the ball as it reaches the net, we can compare it to the height of the net to determine if it clears the net and by how much. - If \(y > 1.07 \mathrm{~m},\) the ball clears the net and the difference is given by \(y - 1.07 \mathrm{~m}\) - If \(y < 1.07 \mathrm{~m},\) the ball doesn't clear the net and the difference is given by \(1.07 \mathrm{~m} - y\) Using the given information and following these steps, we can determine if the ball clears the net and by how much.

Key concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It's all about how things move, focusing on the different types of motion:

• Linear Motion: Movement in a straight line, like a sprinter racing down a track.
• Rotational Motion: Movement around an axis, like a spinning ice skater.
• Projectile Motion: Where our tennis ball comes into play, involving movement in both horizontal and vertical planes.

Understanding kinematics helps us analyze the path of an object, its velocity, and acceleration, which are key to solving problems like whether the tennis ball clears the net.

Horizontal and Vertical Components

In projectile motion, any initial velocity can be broken down into two components: horizontal and vertical. This helps in analyzing the motion separately in each direction.

The horizontal component, \(v_{0x}\), is found by multiplying the initial speed by the cosine of the launch angle. This tells us how fast the object is moving horizontally.

The vertical component, \(v_{0y}\), is calculated using the sine of the launch angle. It indicates the initial speed in the vertical direction.

Understanding these components is crucial because they determine how far and high the projectile will travel.

Trigonometry

Trigonometry is the mathematical tool that enables us to calculate the components of velocity in projectile motion. Using trigonometric functions like sine and cosine, we can easily determine the horizontal and vertical velocities.

• Cosine: Used to calculate the horizontal velocity component, as it's adjacent to the angle in right-angled triangles.
• Sine: Used for the vertical component, as it's opposite the angle.

By applying these functions, we break down a complex motion into simpler components, making it easier to analyze and predict the trajectory.

Free Fall

Free fall refers to the motion of objects under the influence of gravity alone. In the case of a tennis ball, once it leaves the racket, it becomes a freely falling object, subject to gravitational forces.

During free fall, the only force acting on the ball is gravity, causing it to have a constant acceleration downward. This acceleration is standard on Earth, denoted by \(g = 9.81\,\text{m/s}^2\).

Even though the ball also travels horizontally, it behaves like any other freely falling object in the vertical direction, allowing us to predict its motion accurately.

Gravity

Gravity is the force that pulls objects toward the center of the Earth. It's responsible for the acceleration of the tennis ball as it travels in its trajectory.

Here’s what you need to know about gravity in projectile motion:

• Constant Acceleration: Gravity gives the projectile a constant downward acceleration of approximately \(9.81\,\text{m/s}^2\).
• Vertical Influence: While gravity accelerates the ball downward, it doesn't affect horizontal velocity.
• Parabolic Trajectory: Due to gravity’s influence, a projectile follows a curved path.

Understanding gravity’s role is essential for predicting how and where a projectile like a tennis ball will land.