Question

A car of mass $m$ moving with velocity $u$ on a straight road in a straight line, doubles its velocity in time $t$. The power delivered by the engine of a car for doubling the velocity is

• A. $\frac{3mu^2}{2t}$
• B. $\frac{mu^2}{2t}$
• C. $\frac{2mu^2}{t}$
• D. $\frac{3mu^2}{t}$

Hint:

You can use a quick algebraic shortcut: $\Delta K.E. = \frac{1}{2}m(v_f^2 - v_i^2)$.
Substituting $v_f = 2u$ and $v_i = u$ directly gives: $v_f^2 - v_i^2 = 4u^2 - u^2 = 3u^2$.
Plugging this into the power formula yields $P = \frac{3mu^2}{2t}$ in a single step!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A vehicle with a mass of $m$ is travelling down a straight road at an initial speed $u$. Over a continuous time frame $t$, it accelerates until its final speed becomes exactly double its initial value ($2u$). We need to determine the average mechanical power output delivered by the engine during this process.

Step 2: Key Formula or Approach:
1. According to the Work-Energy Theorem, the total net work done ($W$) on an object equals its change in kinetic energy ($\Delta K.E.$): $$W = K.E._f - K.E._i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$ 2. Average power ($P$) is defined as the total mechanical work completed divided by the time interval taken: $$P = \frac{W}{t}$$

Step 3: Detailed Explanation:
Let's define the initial and final velocity conditions: Initial velocity, $v_i = u$ Final velocity, $v_f = 2u$ Calculate the initial kinetic energy of the car: $$K.E._i = \frac{1}{2}mu^2$$ Calculate the final kinetic energy after doubling its speed: $$K.E._f = \frac{1}{2}m(2u)^2 = \frac{1}{2}m(4u^2) = 2mu^2$$ Now, determine the total work done ($W$) using the work-energy relation: $$W = K.E._f - K.E._i = 2mu^2 - \frac{1}{2}mu^2 = \frac{3}{2}mu^2$$ To find the average power delivered by the engine over the duration $t$, divide this work expression by time: $$P = \frac{W}{t} = \frac{\frac{3}{2}mu^2}{t} = \frac{3mu^2}{2t}$$

Step 4: Final Answer:
The power delivered by the engine is $\frac{3mu^2}{2t}$, which matches option (A).