Problem 11 - An Introduction to Physical Science

Question

A sprinter starting from rest on a straight, level track is able to achieve a speed of \(12 \mathrm{m} / \mathrm{s}\) in \(6.0 \mathrm{s}\). What is the sprinter's average acceleration?

Step-by-Step Solution

Answer

The sprinter's average acceleration is \(2 \ \text{m/s}^2\).

Step 1: Identify Known Values

First, list the known values from the problem. The initial speed \( v_0 \) is \( 0 \ \text{m/s}\), as the sprinter starts from rest. The final speed \( v \) is \( 12 \ \text{m/s} \), and the time \( t \) is \( 6.0 \ \text{s} \).

Step 2: Write the Formula for Average Acceleration

The formula for average acceleration \( a \) is given by: \[ a = \frac{v - v_0}{t} \] where \( v \) is the final velocity, \( v_0 \) is the initial velocity, and \( t \) is the time taken to reach the final velocity.

Step 3: Substitute Known Values into the Formula

Substitute the known values into the formula: \[ a = \frac{12 \ \text{m/s} - 0 \ \text{m/s}}{6.0 \ \text{s}} \] This simplifies to: \[ a = \frac{12 \ \text{m/s}}{6.0 \ \text{s}} \]

Step 4: Calculate the Average Acceleration

Perform the division to find the average acceleration: \[ a = 2 \ \text{m/s}^2 \] So, the average acceleration of the sprinter is \( 2 \ \text{m/s}^2 \).

Key Concepts

Understanding Kinematics

Kinematics is the science of motion without considering the forces that cause it. It's about studying how objects move and predicting future movements based on current data. This includes variables like displacement, time, velocity, and acceleration. In this scenario, we are interested in the motion of a sprinter who starts running from rest. Kinematics provides a framework that allows us to describe this motion in terms of time and speed.

Kinematic equations are employed to determine unknown quantities from information we already know. This exercise depicts a fundamental kinematic situation: a sprinter starting from rest and reaching a certain speed. Here, our primary focus is on calculating the sprinter's average acceleration over a given time frame. Understanding kinematics allows us to grasp the relationship between time, velocity changes, and acceleration.

• Displacement: Not in focus here, but usually represents the change in position.
• Acceleration: We focus on how velocity changes over time.

Delving into Velocity

Velocity is a vector quantity, meaning it includes both magnitude and direction. It tells us how fast something is moving and in which direction. In kinematics, velocity is a foundational concept, particularly when evaluating changes over time.

The sprinter's scenario starts with an initial velocity of 0 m/s because the sprinter begins at rest. The final velocity, as given, is 12 m/s. Average acceleration involves the rate of change of this velocity. So, in this case, it looks at how the sprinter's speed increases over the 6-second timeframe.

The equation for average acceleration incorporates both the initial and final velocities. Understanding the change in velocity is key:

• Initial Velocity ( $v_0$ ): This is the velocity at the start, which is 0 m/s.
• Final Velocity ( $v$ ): This is the velocity after time $t$ , here 12 m/s.

Exploring Time in Motion

Time in kinematics scenarios, such as this one, plays a crucial role. It serves as the measurement over which changes in motion, like velocity and acceleration, are evaluated. In the current example, the time span is 6 seconds. It is during this period that the change in velocity and consequently, the acceleration, is measured.

Time is always on the denominator when calculating average acceleration using the formula \[ a = \frac{v - v_0}{t} \],which highlights its role in defining how quickly the speed changes. The duration influences how we interpret an object's acceleration. A large change in velocity over a short time period signifies higher acceleration.

• Evaluation of Time: Helps to discern the change rate in other kinematic quantities like velocity and acceleration.
• Metric of Change: Provides a clear framework to measure and understand dynamical changes in motion parameters.