A test vehicle accelerating from rest to a speed of 10\text{ m/s} requires 50\text{ kJ} of… — Armed Services Vocational Aptitude Battery (ASVAB)

A test vehicle accelerating from rest to a speed of $10\text{ m/s}$ requires $50\text{ kJ}$ of net work. Assuming friction and air resistance are negligible, how much additional work must be done on the vehicle to increase its speed from $10\text{ m/s}$ to $20\text{ m/s}$ ?

A
$50\text{ kJ}$
B
$100\text{ kJ}$
$150\text{ kJ}$ Answer
D
$200\text{ kJ}$

Answer

The additional work required is

150\text{ kJ}

According to the work-energy theorem, net work equals the change in kinetic energy (

W = \Delta KE

). Because kinetic energy is proportional to the square of velocity (

KE = \frac{1}{2}mv^2

), accelerating from rest to

10\text{ m/s}

gives the vehicle

50\text{ kJ}

of kinetic energy. At

20\text{ m/s}

, the speed is doubled, which quadruples the total kinetic energy to

4 \times 50\text{ kJ} = 200\text{ kJ}

. The additional work required to increase the speed from

10\text{ m/s}

20\text{ m/s}

is therefore

200\text{ kJ} - 50\text{ kJ} = 150\text{ kJ}

Step-by-Step Solution

Use the work-energy theorem for the initial acceleration from rest to determine the mass-related factor of the vehicle.

W_1 = \Delta KE_1 = \frac{1}{2}m(v_1^2 - v_0^2) = \frac{1}{2}m(10^2 - 0) = 50m = 50{}000\text{ J} \implies m = 1{}000\text{ kg}

The net work done on an object equals its change in kinetic energy.

Calculate the total kinetic energy of the vehicle at the final speed of

20\text{ m/s}

KE_2 = \frac{1}{2}m v_2^2 = \frac{1}{2}(1{}000\text{ kg})(20\text{ m/s})^2 = \frac{1}{2}(1{}000)(400) = 200{}000\text{ J} = 200\text{ kJ}

Kinetic energy is directly proportional to the square of velocity.

Find the additional work by calculating the change in kinetic energy between

10\text{ m/s}

and

20\text{ m/s}

W_{\text{additional}} = KE_2 - KE_1 = 200\text{ kJ} - 50\text{ kJ} = 150\text{ kJ}

The additional work needed is the difference between final and intermediate energy states.

Key Concept

Work-Energy Theorem and Velocity-Squared Dependence of Kinetic Energy

Estimated Time:1m 30s

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