Problem 14 - College Physics

Question

At a constant speed of \(60 \mathrm{~km} / \mathrm{h}\), an automobile travels \(700 \mathrm{~m}\) along a straight highway that is inclined \(4.0^{\circ}\) to the horizontal. An observer notes only the vertical motion of the car. What is the car's (a) vertical velocity magnitude and (b) vertical travel distance?

Step-by-Step Solution

Answer

Vertical velocity is approximately 1.16 m/s, and vertical travel distance is approximately 48.8 meters.

Step 1: Understanding the setup

The car travels at a constant speed of \(60 \ \mathrm{km/h}\) along a highway inclined at \(4.0^{\circ}\) to the horizontal. We need to find the vertical components of both the velocity and travel distance for this setup.

Step 2: Convert speed to m/s

First, we convert the speed from kilometers per hour to meters per second. We use the conversion factor \(1\ \mathrm{km/h} = \frac{1000}{3600} \ \mathrm{m/s}\). Thus, the speed \(v = 60\ \mathrm{km/h} \approx 60 \times \frac{1000}{3600} \approx 16.67 \ \mathrm{m/s}\).

Step 3: Calculate vertical velocity component

To find the vertical velocity component, we use the sine of the inclination angle. The vertical velocity \(v_{y}\) is calculated as \(v_{y} = v \cdot \sin(\theta)\), where \(v = 16.67\ \mathrm{m/s}\) and \(\theta = 4.0^{\circ}\). Calculate \(v_{y} = 16.67 \cdot \sin(4.0^{\circ})\approx 1.16 \ \mathrm{m/s}\).

Step 4: Calculate total vertical distance

To find the vertical travel distance, we use the sine of the inclination angle and the traveled distance along the incline. If the total distance traveled along the incline is \(d =700\ \mathrm{m}\), then the vertical distance \(d_y = d \cdot \sin(\theta)\). Thus, \(d_y = 700 \cdot \sin(4.0^{\circ})\approx 48.8\ \mathrm{m}\).

Key Concepts

Vertical Velocity

When discussing motion, especially on an inclined plane, vertical velocity becomes an essential component to analyze. Vertical velocity is the rate at which an object moves up or down relative to the vertical axis. It’s a part of the velocity vector split when dealing with inclined situations. To find this component, the object’s total velocity and the inclination angle are key. To determine the vertical velocity, we need to use the sine function of the angle of incline. Mathematically, it is expressed as:- Vertical velocity component, \( v_y = v \cdot \sin(\theta) \).
Here, \( v \) is the total velocity of the object, and \( \theta \) is the angle of inclination relative to the horizontal axis.
- The angle in this example is \( 4.0^{\circ} \) and the speed is converted from kilometers per hour to meters per second for accuracy.
- Understanding this helps in calculating not just how fast an object is going up or down, but also aids in determining its vertical position over time.

Inclined Plane

An inclined plane in physics is a flat surface that is tilted at an angle to the horizontal. It is one of the basic machines in physics that can help reduce the effort of lifting or moving objects. When an object travels on an inclined plane, its motion has both horizontal and vertical components due to the tilt.- In our problem, we are looking at a highway inclined at \( 4.0^{\circ} \) which influences how the car’s motion is separated into different components.
- Inclined planes create motion that can be split into components: the motion along the plane itself and perpendicular to the surface.
- The use of trigonometric functions such as sine and cosine becomes crucial here to separate these components based on the angle of inclination.

Trigonometric Functions

Trigonometric functions are mathematical tools that relate angles to side lengths in triangles. They are extremely useful in physics for resolving different components of vectors, especially on inclined planes.- Sine, cosine and tangent are the primary trigonometric functions used, with sine being particularly useful for vertical components.
- In this context, the sine function allows us to find the vertical component of velocities or distances:
\( \sin(\theta) \) provides the ratio of the opposite side (vertical) to the hypotenuse (inclined distance).
- For example, in our incline problem, using \( \sin(4.0^{\circ}) \) helps determine how much of the car’s motion happens in the vertical direction.
- Mastering these functions aids in accurately dissecting movements in physics problems, ensuring clear and precise calculations.