Question

A rocket ship in free space moves with constant acceleration equal to $9.8\mathrm{m}/{\mathrm{s}}^2.(a)$ If it starts from rest, how long will it take to acquire a speed one-tenth that of light? $(b)$ How far will it travel in so doing? (The speed of light is $3.0\times {10}^8\mathrm{m}/\mathrm{s}$.)

Short answer

The time it will take for the rocket ship to acquire a speed one-tenth that of light can be calculated using the first equation, and the distance it will travel in doing so can be calculated using the second equation.

Step-by-step solution

Calculate Time

Given constant acceleration $a=9.8m/s^2$, final speed that is one-tenth of the speed of light ($v=0.1\times 3.0\times {10}^{8}m/s$), and initial velocity $u=0$ since it starts from rest, we can use the equation $v=u+at$ to find the time it will take to acquire this speed. Solving for $t$ we find $t=(v-u)/a$.

Calculate Distance

Using the time from Step 1, the initial velocity $u=0$, and given acceleration $a=9.8m/s^2$, we can plug these values into the distance equation $s=ut+0.5a{t}^2$ to find the distance the rocket travels.

Key concepts

Constant Acceleration

When an object's velocity changes at a steady rate, we say it is in constant acceleration. This rate of change is crucial in physics, as it allows us to predict an object's motion. In our rocket ship scenario, the acceleration is constant at $9.8\mathrm{m}/{\mathrm{s}}^2$, the same rate of acceleration due to Earth's gravity. This simplicity provides a useful basis to calculate more complex values such as time and distance using kinematic formulas. With constant acceleration, we eliminate the complexities of varying forces and can accurately describe an object's motion in a linear manner over time.

Understanding constant acceleration is fundamental to solving a variety of physics problems, particularly when it comes to motion in space, where external forces are typically minimal to none. Thought of this way, constant acceleration becomes a building block for exploring more advanced aerospace concepts and for simplifying calculations related to motion in a vacuum.

Speed of Light

The speed of light is known to be approximately $3.0\times {10}^8\mathrm{m}/\mathrm{s}$. It's a fundamental constant in physics, representing the maximum speed at which all conventional matter and information in the universe can travel. It is often denoted by the symbol $c$.

In our exercise, the rocket aims to reach a velocity one-tenth of the speed of light, also referred to as $0.1c$. The comparative scale of the speed of light to everyday velocities is tremendous, underscoring the vastness of space and the challenges of space travel. Substantially high speeds, like those approaching the speed of light, bring into play relativistic effects, which are not accounted for in classical mechanics. For our rocket problem, we are functioning within non-relativistic speeds, so we will not delve into these effects, but it’s important to acknowledge the role $c$ plays in both the realms of classical and modern physics.

Kinematic Equations

Kinematic equations are the bread and butter of classical mechanics. They link the variables of motion—distance ($s$), initial velocity ($u$), final velocity ($v$), acceleration ($a$), and time ($t$)—in ways that allow us to solve for one when we know the others. For constant acceleration, these equations provide a reliable framework.

In the problem at hand, we used the first kinematic equation, $v=u+at$, to solve for the time it would take the rocket ship to reach one-tenth of the speed of light. We then used the time found to calculate the distance traveled using the second kinematic equation, $s=ut+0.5a{t}^2$. These equations assume constant acceleration and no external forces acting on the object, making them perfect tools for analyzing the motion of objects in free space, like our rocket.

Free Space Motion

Free space motion refers to the movement of objects in space far from gravitational or atmospheric influences. As such, it provides an almost ideal environment for studying physics principles because there are few external forces to complicate matters. In the context of our exercise, we took advantage of this simplicity to calculate the rocket's time and distance to reach a fraction of the speed of light.

One of the most significant aspects of free space motion is that objects in motion will stay in motion, maintaining their velocity unless acted upon by another force—this is Newton's first law of motion. Since our rocket ship is in free space and is experiencing constant acceleration, we can rely on kinematic equations to describe its motion without the need to factor in variables like air resistance or changing gravitational forces. This isolation makes free space a perfect 'laboratory' for studying motion under constant acceleration.