Question

A baseball pitcher delivers a fastball that crosses the plate at an angle of \(7.25^{\circ}\) relative to the horizontal and a speed of \(88.5 \mathrm{mph}\). The ball (of mass \(0.149 \mathrm{~kg}\) ) is hit back over the head of the pitcher at an angle of \(35.53^{\circ}\) with respect to the horizontal and a speed of \(102.7 \mathrm{mph}\). What is the magnitude of the impulse received by the ball?

Short answer

(Given: mass of the baseball = 0.149 kg) Answer: The magnitude of the impulse received by the baseball is approximately \(7.84 \mathrm{kg \cdot m/s}\).

Step-by-step solution

Convert speeds to SI units

First, we need to convert the given speeds from mph to meters per second (m/s). To perform the conversion, we can use the following formula: Speed (m/s) = Speed (mph) × 0.44704 Initial Speed = \(88.5 \mathrm{mph} \times 0.44704 = 39.56 \mathrm{m/s}\) Final Speed = \(102.7 \mathrm{mph} \times 0.44704 = 45.91 \mathrm{m/s}\)

Find the initial and final momentum components

Next, we need to find the x and y components of the initial and final momenta. To do this, we will use trigonometry. For the initial momentum, we have: Initial momentum in x direction = mass × initial speed × cos(initial angle) Initial momentum in y direction = mass × initial speed × sin(initial angle) For the final momentum, we have: Final momentum in x direction = mass × final speed × cos(final angle) Final momentum in y direction = mass × final speed × sin(final angle) Substitute the given values and calculate the components: Initial momentum in x direction = \(0.149 \mathrm{~kg} \times 39.56 \mathrm{m/s} \times \cos(7.25^{\circ}) = 5.84 \mathrm{kg \cdot m/s}\) Initial momentum in y direction = \(0.149 \mathrm{~kg} \times 39.56 \mathrm{m/s} \times \sin(7.25^{\circ}) = 0.50 \mathrm{kg \cdot m/s}\) Final momentum in x direction = \(0.149 \mathrm{~kg} \times 45.91 \mathrm{m/s} \times \cos(35.53^{\circ}) = 5.68 \mathrm{kg \cdot m/s}\) Final momentum in y direction = \(0.149 \mathrm{~kg} \times 45.91 \mathrm{m/s} \times \sin(35.53^{\circ}) = 8.33 \mathrm{kg \cdot m/s}\)

Calculate the change in momentum

Find the change in the x and y components of the momentum: Change in momentum in x direction = Final momentum in x direction - Initial momentum in x direction Change in momentum in y direction = Final momentum in y direction - Initial momentum in y direction Change in momentum in x direction = \(5.68 \mathrm{kg \cdot m/s} - 5.84 \mathrm{kg \cdot m/s} = -0.16 \mathrm{kg \cdot m/s}\) Change in momentum in y direction = \(8.33 \mathrm{kg \cdot m/s} - 0.50 \mathrm{kg \cdot m/s} = 7.83 \mathrm{kg \cdot m/s}\)

Find the magnitude of the impulse

Finally, find the magnitude of the impulse using the Pythagorean theorem: Impulse magnitude = \(\sqrt{(Change~in~momentum~in~x~direction)^2 + (Change~in~momentum~in~y~direction)^2}\) Impulse magnitude = \(\sqrt{(-0.16 \mathrm{kg \cdot m/s})^2 + (7.83 \mathrm{kg \cdot m/s})^2} = 7.84 \mathrm{kg \cdot m/s}\) The magnitude of the impulse received by the ball is approximately \(7.84 \mathrm{kg \cdot m/s}\).

Key concepts

Conservation of Momentum

Impulse in physics is intimately linked to the principle known as the conservation of momentum. According to this fundamental concept, the total momentum of a closed system remains constant if no external forces act upon it. Momentum is defined as the product of an object's mass and its velocity, and it is a vector quantity, meaning that it has both magnitude and direction.

In the context of our exercise with the baseball pitcher, while the ball experiences a change in momentum as a result of the bat's impact, the combined momentum of the ball and the bat before and after the collision should theoretically remain the same. However, in this scenario, we simplify and focus on the ball's change in momentum to calculate the impulse delivered. Understanding conservation of momentum aids in grasping why the impulse—the cause of the momentum change—is essential for predicting motion in collisions and interactions.

Change in Momentum

Change in momentum is central to the calculation of impulse. When a force is applied to an object, such as the hit from the bat to the baseball, this results in a change of the object's momentum. Mathematically, impulse (\textbf{J}) can be represented as the product of the average force (\textbf{F}) over the time (\textbf{t}) the force was applied: \( \textbf{J} = \textbf{F} \times \textbf{t} \). But equally, impulse is the change in momentum of the object itself, from before the force was applied to after.

In our baseball example, the ball's momentum changes not just in magnitude, but also in direction due to the hit. Therefore, we calculate the change in momentum along each direction (x and y components) and then combine these to find the total change in momentum vector, from which we derive the magnitude of impulse.

Trigonometry in Physics

Trigonometry plays a crucial role in physics, particularly when analyzing vectors and their components. In our baseball exercise, trigonometry allows us to resolve the momentum of the baseball, which is a vector, into its horizontal (x) and vertical (y) components. By using sine and cosine functions, we can accurately determine these components based on the angle of the ball's trajectory.

For instance, to find the components of the initial and final momentum, we multiply the ball's mass and speed by the cosine of the angle for the x-direction and by the sine of the angle for the y-direction. This usage of trigonometry is essential for a wide array of physics problems involving forces, motion, waves, and other phenomena where directional components are involved.

Converting Units

In physics, it's often necessary to convert units to perform calculations correctly. Many physics equations require that we use standard International System of Units (SI), such as meters per second (m/s) for velocity. In our problem, the speeds were initially given in miles per hour (mph), so for compatibility with SI units and to use them in our equations, we converted them to m/s using the conversion factor 0.44704.

Learning to convert units properly is an essential skill, not just for solving physics problems but also for understanding and applying scientific measurements in real-world situations. It's important to remember that the choice of units can affect the outcome and interpretation of a problem, making conversions a fundamental step in the problem-solving process.