Question

A pellet gun fires ten \(2.14\) -g pellets per second with a speed of \(483 \mathrm{~m} / \mathrm{s}\). The pellets are stopped by a rigid wall. (a) Find the momentum of each pellet. (b) Calculate the average force exerted by the stream of pellets on the wall. \((c)\) If each pellet is in contact with the wall for \(1.25 \mathrm{~ms}\), what is the average force exerted on the wall by each pellet while in contact? Why is this so different from \((b)\) ?

Short answer

(a) The momentum of each pellet is \(1.03402 kg*m/s\). (b) The average force exerted by the stream of pellets on the wall is \(10.3402 N\). (c) The average force exerted on the wall by each pellet while in contact is \(827.216 N\). The large difference between this force and the force in part (b) is due to the fact that each pellet only exerts a force during a tiny time interval, giving a much larger average during this time.

Step-by-step solution

Calculate the Momentum of Each Pellet

Momentum (p) can be calculated by multiplying the mass (m) of the pellet by its velocity (v), or \(p = m \cdot v\). Given that the mass of each pellet is \(2.14g = 0.00214kg\) and the velocity is \(483m/s\), we substitute these values into the formula to find the momentum.

Calculate the Average Force Exerted by the Stream of Pellets

Force (F) is the rate of change of momentum, which can be calculated by dividing the total momentum by the total time (F=p/t). Because ten pellets are fired each second, the total momentum each second is the momentum of each pellet multiplied by ten. Substituting these values into the formula will give the average force exerted on the wall by the stream of pellets.

Calculate the Average Force Exerted by Each Pellet

We already know the momentum of each pellet from Step 1, but the time each pellet is in contact with the wall is now \(1.25ms = 0.00125s\). This time is used in the force calculation (F = p/t) instead of one second. Substituting these values into this formula will give the average force each pellet exerts on the wall. The difference from the force in Step 2 arises from the fact that each pellet only exerts a force during a small time interval, resulting in a much larger average force during this short period.

Key concepts

Linear Momentum

Linear momentum is a fundamental concept in physics that describes the motion of an object. It is a vector quantity, which means it has both a magnitude and a direction, and is given by the product of an object's mass and its velocity. The formula for linear momentum \( p \) can be expressed as \( p = m \cdot v \) where \( m \) is the mass and \( v \) is the velocity of the object.

In our exercise, the momentum of each pellet fired by the pellet gun is determined using this formula. Each pellet has a mass of \( 0.00214 \) kg and velocity of \( 483 \) m/s. Once you multiply these values, you can find the linear momentum for each pellet. Understanding the momentum of an object is crucial as it helps to predict the outcomes in collisions and plays a vital role in the conservation of momentum principle.

Average Force

The concept of force is well-known to most students, but average force may need some explanation. Average force is the constant force that would produce the same effect on an object's motion as the actual force exerted over a given time period. The formula for average force \( F \) when considering momentum is \( F = \dfrac{\Delta p}{\Delta t} \) where \( \Delta p \) is the change in momentum and \( \Delta t \) is the change in time.

In our exercise, the average force exerted by a stream of pellets can be calculated by multiplying the momentum of one pellet by the number of pellets fired per second to obtain the total momentum change per second. Using this method, we determine the continuous impact on the wall by the multiple pellets over time.

Rate of Change of Momentum

The rate of change of momentum is a way to understand how the momentum of an object is altered over time, and it is directly related to the force applied. Newton's second law of motion states that the force applied to an object is equal to the rate of change of its momentum. This is especially important when considering objects in collision or coming to a stop, like the pellets in the provided exercise.

In the context of our exercise, when a pellet hits the wall, its momentum changes from the initial momentum it had before impact to zero, since the wall stops the pellet. The rate at which this momentum changes is what we calculate to find the average force exerted during the short time the pellet is in contact with the wall.

Impact Time

Impact time refers to the duration over which a collision or contact between two objects occurs. It's an essential factor in force calculations, as it helps establish the average force exerted during the impact. A shorter impact time generally means a higher force if the change in momentum is the same.

In the exercise, each pellet is in contact with the wall for a mere \( 1.25 \) milliseconds, which is a very short time. The average force exerted by each pellet on the wall is considerably higher during this brief impact time than the average force calculated over one second. This illustrates how force can vary greatly depending on the duration of the impact, even if the momentum change is identical.