Problem 45 - College Physics

Question

A car coasts (engine off) up a \(30^{\circ}\) grade. If the speed of the car is \(25 \mathrm{~m} / \mathrm{s}\) at the bottom of the grade, what is the distance traveled by the car before it comes to rest?

Step-by-Step Solution

Answer

The car travels approximately 63.774 meters before stopping.

Step 1: Identify the Problem Type

The problem is about motion with a change in elevation and involves gravity and potential energy. We need to find the distance the car travels uphill before stopping, using concepts from energy conservation.

Step 2: Understand Known Values

We know that the initial speed of the car is from a stop at the bottom of the hill where \(v_i = 25 \text{ m/s}\) and the hill is inclined at \(30^{\circ}\). The final velocity \(v_f = 0\) since the car comes to rest.

Step 3: Apply Energy Conservation Principle

Use the conservation of mechanical energy principle, where the initial kinetic energy is equal to the potential energy at the maximum height. The formula is: \( \frac{1}{2} m v_i^2 = mgh \), where \(h\) is the height reached.

Step 4: Simplify the Energy Equation

Since mass \(m\) cancels out on both sides of the equation, we have \( \frac{1}{2} v_i^2 = gh \). Substitute \(g = 9.8 \text{ m/s}^2\) and \(v_i = 25 \text{ m/s}\) to find \(h\).

Step 5: Solve for Height \(h\)

Solve for \(h\):\[ h = \frac{v_i^2}{2g} = \frac{25^2}{2 \times 9.8} = \frac{625}{19.6} \approx 31.887 \text{ m} \]

Step 6: Relate Height to Distance Traveled

The relationship between height \(h\) and the distance traveled along the incline \(d\) can be represented as \(d = \frac{h}{\sin(\theta)}\). With \(\theta = 30^{\circ}\), the sine of the angle is \(0.5\).

Step 7: Calculate the Distance \(d\)

Use \(d = \frac{h}{\sin(30^{\circ})} = \frac{31.887}{0.5} \approx 63.774 \text{ m} \). So, the car travels approximately \(63.774\) meters before it stops.

Key Concepts

Kinetic Energy

Kinetic energy refers to the energy that an object possesses due to its motion. It is a crucial concept in physics, especially when analyzing moving objects like cars, cyclists, or even just a rolling ball. Kinetic energy depends on two factors: the mass of the object and the speed at which it is moving. Mathematically, it is expressed by the formula:

• \( KE = \frac{1}{2} mv^2 \)

Here, \(m\) is the mass and \(v\) is the velocity of the object. This formula shows that the kinetic energy is directly proportional to the mass and increases with the square of the velocity.
This means if the speed of an object doubles, its kinetic energy increases by a factor of four, illustrating the profound impact that velocity has on kinetic energy. In the context of our example, the car starting at the bottom of the inclined plane with a speed of 25 m/s possesses a certain amount of kinetic energy initially. As the car moves upward, this energy gets converted into potential energy, eventually bringing the car to rest.

Potential Energy

Potential energy is the stored energy that an object has due to its position or state. One common type is gravitational potential energy, which depends on the height of an object above the ground and the force of gravity acting on it.
The formula for gravitational potential energy is:

• \( PE = mgh \)

Here, \(m\) represents the mass, \(g\) is the acceleration due to gravity (9.8 m/s²), and \(h\) is the height.
As the car moves up the inclined plane, it gains height and therefore potential energy. This energy comes from the loss of kinetic energy as the car slows down.
In the car's journey up the incline, kinetic energy is converted into gravitational potential energy until the car eventually stops at its highest point, where all its initial kinetic energy has been transformed into potential energy.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, which can be used to study the effects of forces and energy transformations more easily.
It is one of the six classical simple machines that help us to do work by reducing the force needed to raise objects. In our scenario, it's a 30-degree slope which affects how the car's kinetic energy is converted into potential energy.
Understanding the relationship between the height and the distance traveled along the incline is essential.

• The height \(h\) is related to the distance \(d\) and angle \(\theta\) by \(h = d \cdot \sin(\theta)\).

With a steeper incline, the conversion of kinetic to potential energy occurs more rapidly because the gravitational force component acting parallel to the surface is larger.
Hence, the car comes to a stop sooner compared to a flatter incline. Overall, the inclined plane plays a pivotal role in this exercise by dictating how far and how fast the car will travel until it stops.