Your forehead can withstand a force of about 6.0 kN before fracturing, while your cheekbone can withstand only about 1.3 kN. Suppose a 140 g baseball traveling at 30 m/s strikes your head and stops in 1.5 ms. What is the magnitude of the force that stops the baseball?

Verified step by step guidance

1. Step 1: Identify the given values in the problem. The mass of the baseball is 140 g (convert to kilograms: 0.140 kg), its initial velocity is 30 m/s, its final velocity is 0 m/s (since it stops), and the time taken to stop is 1.5 ms (convert to seconds: 0.0015 s).
2. Step 2: Use the impulse-momentum theorem, which states that the impulse (force × time) is equal to the change in momentum. The formula is: $Ft=m(v$_f-v_i), where F is the force, t is the time, m is the mass, v_f is the final velocity, and v_i is the initial velocity.
3. Step 3: Rearrange the formula to solve for the force: $F=m(v$_f-v_i)/t. Substitute the values: m = 0.140 kg, v_f = 0 m/s, v_i = 30 m/s, and t = 0.0015 s.
4. Step 4: Calculate the change in velocity: $v$_f-v_i=(0-30)=-30 m/s. Then substitute this into the formula for force.
5. Step 5: Perform the division to find the magnitude of the force: $F=(0.140-30)/0.0015$. The result will give the magnitude of the force that stops the baseball.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). This principle is essential for calculating the force exerted on the baseball as it comes to a stop, as it allows us to relate the change in velocity to the force applied over a specific time interval.

Impulse and Momentum

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can be calculated as the product of the average force and the time duration of the force application. Understanding impulse is crucial for determining the force that stops the baseball, as it connects the force exerted to the change in momentum of the baseball.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In this context, kinematics helps us understand the initial velocity of the baseball and how it decelerates to a stop, providing the necessary parameters to calculate the force involved in the collision.