Find the time it takes for a spaceship to reach the speed of light and the distance it travels, given its initial velocity and a time-dependent acceleration.

Spaceship Accelerating to the Speed of Light

Antiderivatives core

A spaceship is moving in a straight line at $36,000$ miles per second. Suddenly it accelerates at $a = 18t \text{ mi/s}^2$. Assume that the speed of light is $180,000 \text{ mi/s}$ and that relativistic effects do not influence the velocity of the space ship. How long does it take the ship to reach the speed of light and how far does it travel during that time?

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What you need first

Basic integration of polynomials
Solving quadratic equations
Simplifying square roots
Understanding the relationship between position, velocity, and acceleration

What this problem tests

Tests the ability to set up and solve initial value problems in the context of rectilinear motion.

This is a standard application of integration in a first-semester calculus course, demonstrating how to recover velocity and position from acceleration.

Spaceship Accelerating to the Speed of Light

What you need first

What this problem tests

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