An industrial catapult uses a compressed mechanical spring that stores 1{}600\text{ J} of … — Armed Services Vocational Aptitude Battery (ASVAB)

An industrial catapult uses a compressed mechanical spring that stores $1{}600\text{ J}$ of elastic potential energy. Upon release, all of this stored energy is converted entirely into kinetic energy to propel an $8\text{ kg}$ projectile horizontally across a frictionless launch track. What is the velocity of the projectile immediately after launch?

A
$10\text{ m/s}$
$20\text{ m/s}$ Answer
C
$200\text{ m/s}$
D
$400\text{ m/s}$

Answer

20\text{ m/s}

According to the principle of conservation of mechanical energy, the

1{}600\text{ J}

of stored elastic potential energy is completely converted into kinetic energy (

KE = 1{}600\text{ J}

). Applying the kinetic energy formula

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

with

m = 8\text{ kg}

gives

1{}600 = \frac{1}{2}(8)v^2 = 4v^2

. Solving for velocity gives

v^2 = 400

, and taking the square root yields

v = 20\text{ m/s}

Step-by-Step Solution

Apply the law of conservation of mechanical energy to equate potential and kinetic energy.

KE = PE_{\text{elastic}} = 1{}600\text{ J}

Since the launch track is frictionless and energy is fully converted, initial potential energy equals final kinetic energy.

Substitute the known mass and kinetic energy into the kinetic energy equation.

1{}600 = \frac{1}{2} (8\text{ kg}) v^2

The standard formula for kinetic energy is

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

Simplify the equation to solve for

v^2

4v^2 = 1{}600 \implies v^2 = 400\text{ m}^2/\text{s}^2

Dividing both sides by 4 isolates the squared velocity term.

Take the square root of both sides to determine velocity.

v = \sqrt{400} = 20\text{ m/s}

The final velocity is the positive square root of

v^2

Key Concept

Conservation of Mechanical Energy and Kinetic Energy Calculation

Alternative Method

Express velocity algebraically first:

v = \sqrt{\frac{2E}{m}} = \sqrt{\frac{2 \times 1{}600}{8}} = \sqrt{\frac{3{}200}{8}} = \sqrt{400} = 20\text{ m/s}

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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