Problem 26 - College Physics

Question

In an emergency stop to avoid an accident, a shoulder-strap seatbelt holds a 60 -kg passenger in place. If the car was initially traveling at \(90 \mathrm{~km} / \mathrm{h}\) and came to a stop in \(5.5 \mathrm{~s}\) along a straight, level road, what was the average force applied to the passenger by the seatbelt?

Step-by-Step Solution

Answer

The average force applied by the seatbelt is 272.7 N.

Step 1: Convert Speed to Meters per Second

First, we need to convert the speed of the car from kilometers per hour (km/h) to meters per second (m/s). The car's initial speed is 90 km/h. We use the conversion factor \(1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}}\):\[90 \text{ km/h} = 90 \times \frac{1000}{3600} \text{ m/s} = 25 \text{ m/s}\]

Step 2: Calculate the Acceleration

With the initial speed known, we calculate the acceleration (or deceleration, since the car is stopping) using the formula for constant acceleration:\[a = \frac{v_f - v_i}{t}\]Where \(v_f = 0 \text{ m/s}\) (final speed), \(v_i = 25 \text{ m/s}\) (initial speed), and \(t = 5.5 \text{ s}\) is the time.\[a = \frac{0 - 25}{5.5} = -4.545 \, \text{m/s}^2\]

Step 3: Calculate the Force Applied

The force exerted by the seatbelt can be found using Newton's second law of motion, \(F = ma\), where \(m = 60 \text{ kg}\) is the mass of the passenger and \(a = -4.545 \text{ m/s}^2\) is the acceleration:\[F = 60 \times (-4.545) = -272.7 \, \text{N}\]The negative sign indicates the force is in the opposite direction of the initial motion (it is a deceleration force).

Step 4: Interpret the Force Direction

The force magnitude is 272.7 N. Even though the calculated force is negative, this simply reflects the direction of the force as it acts opposite the direction of motion. As a result, the force is understood as 272.7 N acting backwards.

Key Concepts

Constant Acceleration

Acceleration occurs when there is a change in velocity. When we talk about **constant acceleration**, it means that the change in velocity happens at a steady rate. In our exercise, the car decelerates at a constant rate as it comes to a stop. This is important because it allows us to use simple formulas for calculations, like the one for acceleration:

• \( a = \frac{v_f - v_i}{t} \)

Here, \(v_f\) is the final velocity (0 m/s in this case), \(v_i\) is the initial velocity (25 m/s, converted from 90 km/h), and \(t\) is the total time taken (5.5 s) for the car to stop.
Applying this formula lets us find out the steady rate of slowdown, known as the deceleration, which is an essential part of Newton's Second Law. This consistency makes it easier to predict and calculate the forces involved.

Deceleration

Deceleration is the process of slowing down. It's a type of acceleration, but it's directed opposite to the motion. In the given exercise, the car's initial speed reduces to zero as it stops. This is a clear case of deceleration.
When computing deceleration:

• We look for a negative acceleration value, indicating the car slows instead of speeds up.
• Our negative acceleration in this instance is \(-4.545 \, \text{m/s}^2\), showing the car's reduction in speed.

This clarifies the seatbelt's role in applying force to stabilize the passenger when negative acceleration, or deceleration, occurs. Deceleration ensures safety by minimizing collision impact.

Force Calculation

Calculating the force involves Newton's Second Law, which is crucial in physics. Newton's law states:

• **Force (F) = mass (m) \(\times\) acceleration (a)**

In the exercise, since the car suddenly decelerates, the seatbelt must exert a force to bring the passenger to rest:

• Mass \(m\) is 60 kg (the passenger's weight).
• The acceleration \(a\) calculated is \(-4.545 \, \text{m/s}^2\).

Putting these into the formula gives the force as:

• \( F = 60 \times (-4.545) = -272.7 \, \text{N} \)

The negative sign here indicates the direction of the force is opposite to the initial motion. It's an important indicator of deceleration, showing that the seatbelt pulls back against the passenger's forward momentum.

Kinematics

Kinematics involves studying motion without considering the forces that cause it. It helps us understand how objects move based on velocity, time, and acceleration. In our exercise, using kinematics:

• The initial speed of the car (25 m/s) had to be converted from 90 km/h for accurate calculations.
• The time for the car to stop was 5.5 seconds.
• Consequently, the whole stopping scenario is modeled through these kinematic equations.

Ultimately, kinematics provides a framework to visualize the changing motion of the car as it stops. By determining parameters like initial velocity and time, kinematics plays a vital role in solving real-life problems accurately.