Question

In a baseball game, a batter hits the ball at a height of \(4.60 \mathrm{ft}\) above the ground so that its angle of projection is \(52.0^{\circ}\) to the horizontal. The ball lands in the grandstand, \(39.0 \mathrm{ft}\) up from the bottom; see Fig. 4-38. The grandstand seats slope upward at \(28.0^{\circ}\) with the bottom seats \(358 \mathrm{ft}\) from home plate. Calculate the speed with which the ball left the bat. (Ignore air resistance.)

Short answer

After carrying out these calculations, the initial speed of the ball can be found. This requires an understanding of both the horizontal and vertical components of the initial velocity, the angle of projection, and the distance from home plate to where the ball lands in the grandstand.

Step-by-step solution

Calculate the horizontal distance

First, calculate the horizontal distance \(d\) from home plate to the point where the ball lands using the formula \(d = l \cos \theta\), where \(l = 358 \, \text{ft}\) is the distance from home plate to the bottom seats and \(\theta = 28.0^{\circ}\) is the angle of upward slope. This gives \(d = 358 \, \text{ft} \cos(28.0^{\circ}) = 316 \, \text{ft}\).

Calculate time of flight

The time of flight \(t\) can be found from the equation of motion in vertical direction: \(h = u_y t - \frac{1}{2} g t^2\), where \(h = 35 \, \text{ft}\) is the vertical distance from the point of projection to the point of landing, \(u_y = u \sin \alpha\) is the initial vertical velocity component and \(\alpha = 52.0^{\circ}\) is the angle of projection. Treating this equation as a quadratic equation \(at^2 + bt + c = 0\), where \(a = - \frac{1}{2} g\), \(b = u \sin \alpha\) and \(c = - h\), we can then use the quadratic formula to find the relevant root which gives the time of flight.

Calculate initial speed

To calculate the initial speed \(u\) of the ball, we can use the understand that horizontal velocity remains constant during the flight. The horizontal velocity \(u_x = u \cos \alpha\) where \(\alpha = 52.0^{\circ}\) is the angle of projection. By using the equation \(d = u_x t\), where \(d = 315 \, \text{ft}\) is the horizontal distance calculated in Step 1 and \(t\) is the time of flight obtained in Step 2, we can solve for \(u\) to find the speed with which the ball left the bat.

Key concepts

Kinematic Equations

When we examine the motion of a projectile, such as a baseball, we rely on kinematic equations to describe its path. These equations link variables like time, velocity, acceleration, and displacement. Understanding these fundamental relationships is key to solving any projectile motion problem.

For vertical motion, we typically use the equations:

• Initial velocity component in y-direction: \( u_y = u \sin(\alpha) \)
• Displacement in y-direction: \( y = u_y t - \frac{1}{2} g t^2 \)

Here, \( u \) is the initial velocity, \( \alpha \) the angle of projection, \( t \) the time, and \( g \) the acceleration due to gravity (approximately \( 9.81m/s^2 \) or \( 32ft/s^2 \) on Earth). The horizontal motion, unaffected by gravity in our air-resistance-free scenario, uses the equation: \( x = u_x t \), where \( u_x \) is the horizontal component of the initial velocity.

Angle of Projection

The angle at which a projectile is launched, known as the angle of projection, heavily influences its trajectory. A larger angle tends to increase the vertical distance while possibly decreasing the horizontal distance covered, depending on other factors such as initial speed.

In the case of our baseball problem, the angle of projection is given as \(52.0^\circ\), which means we must break down the initial velocity into its vertical and horizontal components using trigonometric functions like sine and cosine.

Initial Velocity

The initial velocity, or the speed at which the object begins its journey, determines how far and how high the projectile will go. Bearing in mind the angle of projection, we can determine the separate components of this velocity as well.

The initial vertical velocity in our baseball example, calculated using the sine function, is vital for understanding how high the ball will travel before succumbing to gravity’s pull.

Time of Flight

Time of flight refers to the duration that the projectile spends airborne. It affects how far the object will travel horizontally during its journey. To calculate it, we often turn to the vertical motion equation and solve for time, because vertical motion is influenced by gravity, providing a solvable quadratic equation.

Finding the time of flight enables us to sync up the horizontal and vertical motion aspects of the projectile's path, offering a complete picture of the trajectory.

Quadratic Formula

The quadratic formula, \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a critical tool for solving quadratic equations that appear in projectile motion problems. It allows us to calculate the unknown variable, which is often time, from the arranged standard form equation \( at^2 + bt + c = 0 \).

In our baseball scenario, we translate the vertical motion equation into this standard form to facilitate the use of the quadratic formula and solve for time. Careful interpretation of the formula’s two possible solutions is required to find the physically meaningful time of flight.

Horizontal Distance

The horizontal distance is the measure of how far the projectile travels along the x-axis from its origin. For our purposes, this translates to the distance the baseball moves from the bat to the point it lands in the stands. Since horizontal velocity is constant (ignoring air resistance), the horizontal distance can be found by multiplying the horizontal component of the velocity by the time of flight.

To calculate it accurately, it is crucial to comprehend the underlying trigonometry and kinematics involved in relating time, distance, and velocity.

Vertical Distance

Vertical distance is the altitude change of the projectile from its starting point to its end point. It is determined by the initial velocity's vertical component, acceleration due to gravity, and the time of flight.

In our example, the ball's vertical displacement includes its initial height above the ground and the height at which it lands in the grandstands. This information is used in conjunction with the time of flight to solve for the projectile’s initial speed using the kinematic equations for vertical motion.