Question

A pellet gun fires ten 2.0gpellets per second with a speed of$\mathbf{500}\frac{\mathbf{m}}{\mathbf{s}}$. The pellets are stopped by a rigid wall. What are (a) The magnitude of the momentum of each pellet,(b) The kinetic energy of each pellet, and (c) The magnitude of the average force on the wall from the stream of pellets? (d) If each pellet is in contact with the wall for 0.60 ms, what is the magnitude of the average force on the wall from each pellet during contact? (e) Why is this average force so different from the average force calculated in (c)?

Short answer

1. Magnitude of momentum of each pellet is$1\frac{\mathrm{kg}.\mathrm{s}}{\mathrm{s}}$.
2. Kinetic energy of each pellet is 250 J.
3. Magnitude of average force on the wall from stream of pellets is 10 N.
4. Magnitude of average force if is 1700 N.
5. Contact time is different t = 0.6 sec is 1700 N.

Step-by-step solution

Step 1: Given

1. Number of pellets is, n = 10
2. Mass r of pellets is, m = 2g = 0.002 kg
3. Speed of pellets is,$v=500\frac{\mathrm{m}}{\mathrm{s}}$

Determine the formulas

Use the basic formula of momentum, which is the product of mass and velocity. Then to find kinetic energy, use the basic formula of mass and square of velocity. Then, use the basic formula of the average force, which is ratio of momentum and time.

Formulae are as follow:

$$\begin{gathered} P=mv \\ KE=\frac{1}{2}m{v}^2 \\ F=\frac{P}{t} \end{gathered}$$

Here, P is momentum, m is mass, v is velocity, KE is kinetic energy, F is force and t is time.

(a) Determining the magnitude of momentum of each pellet

To calculate magnitude of momentum of each pellet, use the following formula,

$$\begin{gathered} P=mv \\ P=0.002\times 500 \\ P=1\frac{\mathrm{kg}.\mathrm{m}}{\mathrm{s}} \end{gathered}$$

Hence,the magnitude of momentum of each pellet is $1\frac{\mathrm{kg}.\mathrm{m}}{\mathrm{s}}$.

(b) Determining the kinetic energy of each pellet

Kinetic energy of each pellet,

$$\begin{gathered} KE=\frac{1}{2}\times m\times v^2 \\ KE=0.5\times 0.002\times {500}^2 \\ KE=250\mathrm{J} \end{gathered}$$

Hence, the kinetic energy of each pellet is 250J.

(c) Determining the magnitude of average force on the wall from stream of pellets

Average force on n pallet is,

$$\begin{gathered} F_{avg}=\frac{nP}{t} \\ {F}_{avg}=\frac{10\times 1}{1} \\ {F}_{avg}=10\mathrm{N} \end{gathered}$$

Hence,the magnitude of average force on the wall from stream of pellets is 10N.

(d) Determining the magnitude of average force if t = 0.6 sec

Average force when t = 0.6 s:

$F_{avg}=\frac{P}{t}$

Substitute the values and solve as:

$$\begin{gathered} F_{avg}=\frac{1}{0.6\times {10}^{-3}} \\ {F}_{avg}=1666.67 \end{gathered}$$

In two significant figures,

$F_{avg}=1700\mathrm{N}$

Hence,the magnitude of average force if t = 0.6 sec is 1700 N.

(e) Determine the average force

Average forces are different because the contact time is different. For case 1, it isand for case 2, it is 0,6 s.

Therefore, the momentum, K.E and average force on an object can be found using the corresponding formulae.