Problem 33 - Fundamentals of Physics

Question

A car traveling \(56.0 \mathrm{~km} / \mathrm{h}\) is \(24.0 \mathrm{~m}\) from a barrier when the driver slams on the brakes. The car hits the barrier \(2.00 \mathrm{~s}\) later. (a) What is the magnitude of the car's constant acceleration before impact? (b) How fast is the car traveling at impact?

Step-by-Step Solution

Answer

(a) Acceleration is -3.55 m/s². (b) Impact speed is 8.45 m/s.

Step 1: Convert initial speed to meters per second

To solve this, the speed needs to be in meters per second. Convert the initial speed of the car from 56.0 km/h to m/s. Use the conversion 1 km/h = 0.27778 m/s. \[ v_0 = 56.0 \times 0.27778 = 15.55 \text{ m/s} \]

Step 2: Identify known variables and equation

We have the initial velocity \( v_0 = 15.55 \text{ m/s} \), the time \( t = 2.00 \text{ s} \), and the distance \( d = 24.0 \text{ m} \). We need to find the acceleration \( a \). The kinematic equation connecting these variables is \[ d = v_0 t + \frac{1}{2} a t^2 \].

Step 3: Solve for acceleration

Substitute the known values into the equation and solve for \( a \): \[ 24.0 = 15.55 \times 2.00 + \frac{1}{2} a (2.00)^2 \]. This simplifies to \[ 24.0 = 31.10 + 2a \]. Thus, \[ 2a = 24.0 - 31.10 \rightarrow 2a = -7.10 \rightarrow a = -3.55 \text{ m/s}^2 \].

Step 4: Determine the final speed at impact

To find the final velocity at impact \( v \), use the formula \( v = v_0 + at \), where \( v_0 = 15.55 \text{ m/s} \), \( a = -3.55 \text{ m/s}^2 \), and \( t = 2.00 \text{ s} \). Calculate \[ v = 15.55 + (-3.55) \times 2.00 = 15.55 - 7.10 = 8.45 \text{ m/s} \].

Key Concepts

Constant Acceleration

The concept of constant acceleration is central in understanding the motion of an object when forces are acting upon it. Constant acceleration means that the object's velocity changes at a steady rate over time.

In this exercise, constant acceleration refers to the car's acceleration as it slows down before hitting the barrier. This is different from the car speeding up or moving at a constant speed. The forces acting such as the brakes cause the car to slow down in a uniform manner.

Constant acceleration can be represented using the formula: \[ a = \frac{{\Delta v}}{{\Delta t}} \]Where:

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

In the given problem, you are given the time it takes the car to hit the barrier and the distance traveled. By using these variables alongside the initial velocity, you are able to calculate the car's acceleration. Negative acceleration indicates the car is slowing down.

Initial Velocity

Initial velocity, often denoted by \( v_0 \), is the speed at which an object begins its motion. It plays a crucial role in kinematic equations, as it helps determine how an object will move over time.

In the example, the car's initial velocity is 56 km/h, which we convert into meters per second as 15.55 m/s using the conversion factor of 1 km/h = 0.27778 m/s. It's important to convert into m/s when working with standard kinematic equations in physics, as they typically use the metric system.

Knowing the initial velocity allows you to predict how the car will move when other factors, such as acceleration and time, are factored in. This value is critical for further calculations, like finding the car's acceleration or its velocity at impact.

Final Velocity

The final velocity, denoted as \( v \), is the speed of an object at the end of a period of time. To find this, you can use the initial velocity and account for any changes caused by acceleration.

The formula to calculate final velocity is:\[ v = v_0 + at \]Where:

• \( v \) is the final velocity
• \( v_0 \) is the initial velocity
• \( a \) is the acceleration
• \( t \) is the time period

By substituting the values \( v_0 = 15.55 \text{ m/s} \), \( a = -3.55 \text{ m/s}^2 \), and \( t = 2.00 \text{ s} \) into the formula, you determine that the car's speed just before impact is 8.45 m/s. This measure of the car's velocity aids in understanding how much the car's speed was reduced due to braking.

Impact Speed

Impact speed reflects the velocity of an object at the precise moment of collision. This speed is crucial in understanding the dynamics of impacts and the potential damage caused.

From previous calculations, the car impacts the barrier at 8.45 m/s. Knowing the impact speed helps infer the severity of the collision and assess necessary safety measures.

Impact speed is determined by various factors including initial velocity and acceleration. By analyzing these, you can establish how forces applied over time altered the object's speed from start to the point of impact.