Question

A runner of mass $56.1\mathrm{kg}$ starts from rest and accelerates with a constant acceleration of $1.23\mathrm{m}/{\mathrm{s}}^2$ until she reaches a velocity of $5.10\mathrm{m}/\mathrm{s}$. She then continues running at this constant velocity. How long does the runner take to travel $173\mathrm{m}?$

Short answer

Answer: To find the total time taken by the runner to travel 173 meters, follow these steps: 1. Calculate the time for the acceleration phase using $t=\frac{v-u}{a}$. 2. Calculate the distance traveled during acceleration using $s=ut+\frac{1}{2}a{t}^2$. 3. Calculate the remaining distance to be covered by subtracting the distance traveled during acceleration from 173 meters. 4. Calculate the time for the constant velocity phase using $t=\frac{s}{v}$. 5. Add the time spent during the acceleration and constant velocity phases to find the total time required for the runner to travel 173 meters.

Step-by-step solution

Calculate the time for the acceleration phase

First, we need to find how long the runner takes to reach the constant velocity of $5.10\mathrm{m}/\mathrm{s}$. We can use the equation $v=u+at$, where $v$ is the final velocity, $u$ is the initial velocity (which is $0$ since the runner starts from rest), $a$ is the acceleration, and $t$ is the time: $$t=\frac{v-u}{a}$$ Plugging in values: $v=5.10\mathrm{m}/\mathrm{s},u=0\mathrm{m}/\mathrm{s}$ and $a=1.23\mathrm{m}/{\mathrm{s}}^2$, we'll get the time for the acceleration phase.

Calculate the distance traveled during acceleration

Next, we need to find the distance traveled during the acceleration phase. We'll use the equation $s=ut+\frac{1}{2}a{t}^2$, where $s$ is the distance traveled, $u$ is the initial velocity, $a$ is the acceleration, and $t$ is the time found in Step 1. Plugging in values: $u=0\mathrm{m}/\mathrm{s},a=1.23\mathrm{m}/{\mathrm{s}}^2$, and $t$ from Step 1, we'll find the distance traveled during acceleration.

Calculate the distance remaining to be covered

Now, we need to find the remaining distance the runner needs to cover before reaching $173\mathrm{m}$. We can do this by subtracting the distance traveled during acceleration (found in Step 2) from $173\mathrm{m}$.

Calculate the time for the constant velocity phase

Now, we'll find the time required to cover the remaining distance at a constant velocity of $5.10\mathrm{m}/\mathrm{s}$. We can use the equation $t=\frac{s}{v}$, where $t$ is the time, $s$ is the distance remaining (found in Step 3), and $v$ is the constant velocity.

Calculate the total time the runner takes to travel $173\mathrm{m}$

Finally, we'll add the time spent during the acceleration phase (Step 1) and the time spent during the constant velocity phase (Step 4) to find the total time required for the runner to travel $173\mathrm{m}$.

Key concepts

Constant Acceleration

Understanding constant acceleration is crucial in solving problems related to motion. An object experiences constant acceleration when its velocity changes at a uniform rate over time. This means the object's speed increases by the same amount every second. For example, in the exercise, the runner starts from rest and changes her velocity uniformly due to constant acceleration.

This concept is described by the equation:

• $v=u+at$

Where:

• $v$ is the final velocity.
• $u$ is the initial velocity (which is 0 if starting from rest).
• $a$ is the acceleration.
• $t$ is the time taken to reach the final velocity.

This formula helps us determine how long it will take the runner to reach a certain speed given a specific acceleration. By using this formula, we calculated how long the runner takes to accelerate from a standstill to 5.10 m/s.

Velocity

Velocity, a key concept in kinematics, is the speed of an object in a particular direction. It's important to distinguish velocity from speed because velocity includes direction, making it a vector quantity. In our exercise, once the runner reaches a velocity of 5.10 m/s, she maintains this speed as she continues running.

Velocity is determined by the equation:

• $v=\frac{s}{t}$

Where:

• $v$ is the velocity.
• $s$ is the distance traveled.
• $t$ is the time taken to travel that distance.

Understanding velocity allows us to solve how long it takes to cover a particular distance while maintaining a constant speed. This is used in the exercise to find the duration of time the runner needs to maintain her velocity to cover the remaining distance after accelerating.

Distance-Time Relationship

The relationship between distance and time is central to solving motion problems. It tells us how the distance an object travels changes over time, depending on whether it's accelerating or moving at a constant speed.

In the context of our exercise, two main scenarios involve this relationship:

• During the acceleration phase, the distance is given by the formula:
• $s=ut+\frac{1}{2}a{t}^2$
• During the constant velocity phase, the distance is given by the formula:
• $s=vt$

In these formulas:

• $s$ is the distance traveled.
• For the acceleration phase, initial velocity $u$ is zero, $a$ is the acceleration, and $t$ is the time to reach final velocity.
• For the constant motion phase, $v$ is the constant velocity, and $t$ is the time to cover the remaining distance.

By using these formulas, we can accurately calculate the total distance the runner travels during each phase of her run and thus determine the total time to cover 173 meters.