Problem 30 - College Physics

Question

A \(50-\mathrm{kg}\) driver sits in her car waiting for the traffic light to change. Another car hits her from behind in a head-on, rear-end collision and her car suddenly receives an acceleration of \(16 \mathrm{~m} / \mathrm{s}^{2}\). If all of this takes place in \(0.25 \mathrm{~s},\) (a) what is the impulse on the driver? (b) What is the average force exerted on the driver, and what exerts this force?

Step-by-Step Solution

Answer

(a) The impulse on the driver is 200 Ns. (b) The average force exerted on the driver is 800 N, exerted by the seat.

Step 1: Define Impulse

Impulse is calculated using the formula:\[ J = F \Delta t = m \Delta v \]where \( J \) is the impulse, \( F \) is the force, \( \Delta t \) is the change in time, \( m \) is the mass, and \( \Delta v \) is the change in velocity. We will need to find \( \Delta v \) in the next step.

Step 2: Calculate Change in Velocity (\(\Delta v\))

Since the car accelerates with a constant acceleration, we use the equation:\[ \Delta v = a \Delta t \]where \( a = 16 \mathrm{~m/s}^2 \) and \( \Delta t = 0.25 \mathrm{~s} \). Substituting the given values:\[ \Delta v = 16 \mathrm{~m/s}^2 \times 0.25 \mathrm{~s} = 4 \mathrm{~m/s} \]

Step 3: Calculate Impulse on the Driver (Part a)

Using the impulse formula,\[ J = m \Delta v \]Substitute \( m = 50 \mathrm{~kg} \) and \( \Delta v = 4 \mathrm{~m/s} \):\[ J = 50 \mathrm{~kg} \times 4 \mathrm{~m/s} = 200 \mathrm{~Ns} \]

Step 4: Calculate Average Force (Part b)

Impulse is also defined as the product of force and time: \[ J = F \Delta t \Rightarrow F = \frac{J}{\Delta t} \]We found \( J = 200 \mathrm{~Ns} \) and \( \Delta t = 0.25 \mathrm{~s} \), so:\[ F = \frac{200 \mathrm{~Ns}}{0.25 \mathrm{~s}} = 800 \mathrm{~N} \]

Step 5: Identify What Exerts the Force

In a rear-end collision, the force exerted on the driver is due to the seat pushing forward on the driver, causing her acceleration.

Key Concepts

Newton's Second Law

Newton's Second Law is a cornerstone of physics, relating the net force acting on an object to its mass and the acceleration it experiences. This principle can be expressed mathematically by the equation:\[ F = m imes a \]Where:

• \( F \) is the net force applied to an object
• \( m \) is the mass of the object
• \( a \) is the acceleration of the object

This law tells us that the force needed to accelerate an object is proportional to its mass. As such, a heavier object requires more force to achieve the same acceleration as a lighter object.
In the context of the car collision problem, the car’s acceleration and the driver’s mass allow us to evaluate the force involved, demonstrating how the impact results in significant force acting over a brief period.

Average Force

Average force is particularly important in understanding the effects of collisions and impacts. During such events, the forces involved can change rapidly. Instead of tracking these changes constantly, we use the concept of **average force** to simplify the analysis.
The average force experienced by an object can be computed using the impulse-momentum theorem:\[ F_{avg} = \frac{J}{\Delta t} \]Where:

• \( F_{avg} \) is the average force
• \( J \) is the impulse, representing the change in momentum
• \( \Delta t \) is the time interval over which the force acts

In the case study, the calculations show how even a brief impact, lasting a fraction of a second, can exert a large average force.
Such insights help us in designing safer vehicles and understanding crash dynamics.

Kinematics

Kinematics is the branch of physics that describes the motion of objects, focusing on parameters like position, velocity, and acceleration, without delving into the forces and torques that cause them.
In problems involving motion under constant acceleration, such as the car collision scenario, kinematic equations help us solve for unknown quantities. One of the fundamental equations used is:\[ \Delta v = a \times \Delta t \]Where:

• \( \Delta v \) is the change in velocity
• \( a \) is the acceleration
• \( \Delta t \) is the time interval

By applying this formula, we can determine how the velocity of the driver changes due to the collision, which then aids in calculating the impulse and average force.
Kinematics provides the tools needed to describe motion effectively, laying the foundation for solving complex physical scenarios.