Question

An automatic gun fires 240 bullets per minute with a speed of 360 km/h. If each bullet weighs 20 g, what is the power produced by the gun?

Answer 1

Hint:The rate at which work is done is referred to as power. It is always dependent on the work done. It is defined as the amount of energy that is converted per unit of time. Its International System of Unit is “Watt”. A single watt of power is equal to one joule of work done per second. It is a scalar quantity as it just represents the rate at which a particular work is being done by someone or something. So, the formula of power is P = (Work done)/(Time).

Formula used:
 P(Power) = W(Work done)/sec(Time).
\[P = n \times \dfrac{1}{2} \times \dfrac{{m{v^2}}}{t}\]
where, n= number of bullets fired
\[KE = \dfrac{{m{v^2}}}{2}\]
Where, m= mass of the bullet, v= velocity of the bullet and t= time.

Complete step by step solution:
Given, v = 360 km/h
Rate of bullets fired, n = 240 bullets per minute
Weight of bullet, m = 20g
Converting the values of velocity from Km/h to m/s we get:
\[v = 360\dfrac{{km}}{{hr}} \\ \Rightarrow v = \dfrac{{360 \times 1000}}{{3600}}\dfrac{m}{{\sec }} \\ \Rightarrow v = 100\dfrac{m}{{\sec }} \\ \]
Now, converting the value of m from ‘g’ to ‘kg’, we get:
\[m = 20g \\ \Rightarrow m = 20 \times {10^{ - 3}}kg \\ \]
Now, the power of a gun as mentioned above will be represented by the formula:
P = Total Kinetic Energy of fired bullet/ Time
\[P = n \times \dfrac{1}{2} \times \dfrac{{m{v^2}}}{t}\]
Putting the values of n, m, and v as given in the question and taking time as 1 min or 60 sec, we get:
\[P = \dfrac{{240 \times \dfrac{1}{2} \times \left( {20 \times {{10}^{ - 3}}} \right) \times {{\left( {100} \right)}^2}}}{2} \\ \Rightarrow P= 4 \times 10 \times {10^{ - 3}} \times {10^4} \\ \therefore P= 400\,W \]

Hence, the power generated by the automatic gun is 400 watts.

Note: Always be careful of the units used in this type of question. Convert the units to suitable ones if required to find the answer. Here kinetic energy of the bullet is taken because it is the form of energy that a body has due to its motion. It is directly proportional to the mass of the object and to the square of its velocity.