Problem 58 - College Physics

Question

\(\bullet\) In 1.00 second an automatic paintball gun can fire 15 balls, each with a mass of \(0.113 \mathrm{g},\) at a muzzle velocity of 88.5 \(\mathrm{m} / \mathrm{s}\) . Calculate the average recoil force experienced by the player who's holding the gun.

Step-by-Step Solution

Answer

The average recoil force is approximately 0.15 N.

Step 1: Convert Mass from grams to kilograms

To work with standard SI units, convert the mass of each paintball from grams to kilograms. The mass of each paintball is given as 0.113 g. We know that 1 g = 0.001 kg, so: \[ 0.113 ext{ g} = 0.113 imes 0.001 ext{ kg} = 0.000113 ext{ kg} \]

Step 2: Calculate Momentum Change per Paintball

Momentum change for each paintball is calculated using the formula: \( \Delta p = m \cdot v \), where \( m \) is the mass and \( v \) is the velocity. With \( m = 0.000113 \text{ kg} \) and \( v = 88.5 \text{ m/s} \), compute the momentum change per ball: \[ \Delta p = 0.000113 \times 88.5 = 0.0099955 \text{ kg} \cdot \text{m/s} \]

Step 3: Calculate Total Momentum Change for All Paintballs

The total momentum change for all paintballs in 1 second is the momentum change per ball multiplied by the number of balls. Given 15 balls are fired in 1 second: \[ \Delta p_{total} = 15 \times 0.0099955 = 0.1499325 \text{ kg} \cdot \text{m/s} \]

Step 4: Calculate Average Recoil Force

Use the impulse-momentum theorem which states that the impulse (force times time) equals the change in momentum: \( F \cdot \Delta t = \Delta p \). Here, \( \Delta t = 1 \text{ s} \) and \( \Delta p = 0.1499325 \text{ kg} \cdot \text{m/s} \). Solve for \( F \): \[ F = \frac{\Delta p}{\Delta t} = \frac{0.1499325}{1} = 0.1499325 \text{ N} \]

Step 5: Finalize Answer

The average recoil force experienced by the player is approximately 0.14993 N, due to rounding based on significant figures in initial data.

Key Concepts

Momentum Change

In the world of physics, understanding momentum change is crucial when assessing the motion of objects, especially those that are rapidly moving, like paintballs. Momentum, defined as the product of an object's mass and velocity, quantifies how much "motion" an object has. The change in momentum (\( \Delta p \)) happens when there's a change in either the mass, velocity, or both.
For each paintball fired, momentum change is calculated using the formula \( \Delta p = m \cdot v \), where \( m \) is the mass and \( v \) is the velocity. In our exercise, the mass of one paintball mass is \( 0.000113 \) kg, and its velocity is \( 88.5 \) m/s. Thus, each paintball has a momentum change given as:
\[\Delta p = 0.000113 \times 88.5 = 0.0099955 \ kg \cdot m/s.\]
This momentum change per ball helps in determining how the firing impacts the recoil experienced by the player holding the paintball gun.

Impulse-Momentum Theorem

The impulse-momentum theorem is an essential principle for understanding forces in scenarios like a paintball gun firing. This theorem relates the impulse applied to an object to its change in momentum.
Impulse is the product of force (\( F \)) and the time period (\( \Delta t \)) during which the force acts. The impulse-momentum theorem states:
\[F \cdot \Delta t = \Delta p\]
This equation shows that the impulse (force times duration) equals the change in momentum.
Applying this to our exercise, where the total momentum change for all 15 paintballs is \( 0.1499325 \) kg \cdot m/s over 1 second, we can calculate the average recoil force. With \( \Delta t = 1 \) s and \( \Delta p = 0.1499325 \), the recoil force is given by:
\[F = \frac{\Delta p}{\Delta t} = \frac{0.1499325}{1} = 0.1499325 \ N.\]
Thus, the player experiences an average recoil force of approximately \( 0.14993 \) N.

SI Units Conversion

In physics, consistent units are paramount for accurate calculations. The International System of Units (SI) is the standard, ensuring uniformity globally.
Our exercise begins with the mass of a paintball given in grams. To maintain consistency with SI units, it is vital to convert all measurements to these units.

• For mass, convert grams to kilograms, since the kilogram is the base unit for mass in the SI system. Knowing \( 1 \) gram equals \( 0.001 \) kilograms, convert \( 0.113 \) grams to kilograms:
\[0.113 \text{ g} = 0.113 \times 0.001 = 0.000113 \text{ kg}.\]

Converting units correctly ensures that subsequent calculations, like the momentum change and force, are accurate. This consistency helps prevent errors and enhances the reliability of scientific computations.