The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than $250$ m/s². If you are in an automobile accident with an initial speed of $105$ km/h ($65$ mi/h) and are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

Verified step by step guidance

1. First, convert the initial speed from kilometers per hour to meters per second. Use the conversion factor: 1 km/h = 0.27778 m/s. Therefore, 105 km/h is equivalent to 105 * 0.27778 m/s.
2. Next, identify the final speed after the stop, which is 0 m/s since the car comes to a complete stop.
3. Use the formula for acceleration: \( a = \frac{v^2 - u^2}{2s} \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( s \) is the stopping distance.
4. Rearrange the formula to solve for the stopping distance \( s \): \( s = \frac{v^2 - u^2}{2a} \). Substitute \( v = 0 \), \( u \) as the converted initial speed, and \( a = 250 \) m/s².
5. Calculate the stopping distance \( s \) using the values substituted into the rearranged formula. This will give you the minimum distance over which the airbag must stop you to ensure survival.

Key Concepts

Acceleration

Acceleration is the rate of change of velocity of an object. It is a vector quantity, having both magnitude and direction, and is measured in meters per second squared (m/s²). In the context of the question, understanding acceleration is crucial to determine the deceleration needed to safely stop a person during a crash.

Kinematics Equations

Kinematics equations describe the motion of objects without considering the forces that cause the motion. They relate displacement, initial velocity, final velocity, acceleration, and time. For this problem, the equation v² = u² + 2as can be used to find the stopping distance, where v is final velocity, u is initial velocity, a is acceleration, and s is displacement.

Conversion of Units

Conversion of units is essential for ensuring consistency in calculations. In this problem, the initial speed is given in kilometers per hour, which must be converted to meters per second to match the units of acceleration. This involves multiplying the speed by a conversion factor of 1000/3600 to convert km/h to m/s.