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Chapter 4: Problem 95

At the cntrance to Ambrone Channel at New York hubor, thc maximum tidal carrent has a velocity of \(4.2 \mathrm{~km} / \mathrm{h}\) in a dircetion \(20^{\circ}\) south of eact. What is the component of this velocity] in the east direction? In the north direction?

Short Answer

Expert verified
East component: 3.95 km/h, North component: -1.44 km/h (southward).

Step by step solution

01

Identify Components and Angle

The velocity vector is given as having a magnitude of \( 4.2 \text{ km/h} \) and a direction \( 20^{\circ} \) south of east. We will resolve this vector into its east and north components.
02

Determine the East Component

The east component of the velocity is given by the formula:\[V_{\text{east}} = V \cdot \cos(\theta)\]where \( V = 4.2 \text{ km/h} \) and \( \theta = 20^{\circ} \).Calculating:\[V_{\text{east}} = 4.2 \cdot \cos(20^{\circ}) \approx 4.2 \cdot 0.9397 \approx 3.95 \text{ km/h}\]
03

Determine the North Component

The direction is south of east, so the northward component is actually negative since it represents southward motion. We find it using:\[V_{\text{north}} = V \cdot \sin(\theta)\]Calculating:\[V_{\text{north}} = 4.2 \cdot \sin(20^{\circ}) \approx 4.2 \cdot 0.3420 \approx 1.44 \text{ km/h}\]Since it is directed south, it should be considered as negative: \(-1.44 \text{ km/h}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometry in Physics
Using trigonometry in physics helps resolve vectors into components, which is essential for understanding motion and forces. By applying trigonometric concepts, one can determine the influence of a vector in specific directions. This concept is widely used in problems like finding velocities in different directions, much like this exercise.

  • Vectors and Trigonometry: A vector has both magnitude and direction, such as a force or velocity. When it is not aligned with a coordinate axis, like our exercise, trigonometry is used to find out how much of it acts in each direction (east and north in this case).
  • Coordinate Axes: By resolving vectors along coordinate axes using sine and cosine functions, we grasp how different forces or velocities combine or oppose each other.

Using the formulas for sine and cosine, physics often simplifies real-world situations, allowing us to predict and calculate motions accurately. This is true in nautical navigation, aerospace dynamics, or sports physics.
Velocity Components
Velocity components are the projections of the velocity vector along the coordinate axes. For example, in our exercise, we break down the velocity into its east and north components. This is crucial for understanding how different parts of a velocity vector influence movement.

  • East Component: This expresses the horizontal influence of the vector. It is calculated using the cosine function, which represents the adjacent side of the angle in trigonometry.
  • North Component: This shows the vertical influence and uses the sine function, representing the opposite side in a right triangle.

Despite the naming, remember that in our situation, the north component has a negative value because the motion is "south of east." Understanding these components helps in solving problems in physics where multiple directional influences operate concurrently.
Angle Resolution in Vectors
Angle resolution in vectors involves breaking a vector into two perpendicular components, using angles to guide the process. In our example, this means using the given angle south of east to determine how the vector moves both east and south.

  • The Angle: Given as 20 degrees south of east, this angle tells us the vector's inclination from a purely horizontal (eastward) path.
  • Decomposition: By resolving the vector at this angle, trigonometric functions (sine and cosine) help find exact component values for motion east and south.

Understanding angle resolution is key to accurately describing and managing real-world forces. It applies from navigating ships to analyzing forces on a structure or studying planetary motion. In each, the correct breakdown of angles provides insight into actual effects a force or direction has within a system.

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Most popular questions from this chapter

In an airport, a moving wallkway has a rpeed of \(1.5 \mathrm{~m} / \mathrm{s}\) relative to the ground. What is the speed, relative to the ground, of a pausenger running forwand on this wallkway at \(4.0 \mathrm{~m} / \mathrm{si}\) ? What is the epeed, relative to the groend, of a passenger running bachward on this walkway at \(4.0 \mathrm{~m} / \mathrm{s}\) ?

A grin on the sbore (at sea level) fures a shet at a ship which is heading dircetly towand the gun at a speed of 40 lun \(\mathrm{h}\). At the intant of firing, the d茅tance to the ship is \(15000 \mathrm{~m}\). The rrupele speed rf the shot is \(700 \mathrm{~m} / \mathrm{s}\). Prctend that there is no, air resistance. (a) What is the reguired elevaticn angle for the gun? Ascume \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) (b) What is the timc interval betwees fininge and impact?

'Two foothall players are initially \(15 \mathrm{~m}\) apart. The firot player (a rerciver) runn perpendicular to the line joining the two players at a contant ipeed of \(8.0 \mathrm{~m} / \mathrm{s}\). Affer two ticconds, the wecond player (the quarterback) thrims the hall with a horimontal omponcnt vclocity of \(20 \mathrm{~m} / \mathrm{s}\). In what hotizontal directicn and with what vertical launch ungle should the quarterback throw so that the hull reaches the same -pot the receivcr will be? At what time will the ball be caught?

A blinp motoring at a constant altirude tus a velocity component of \(15 \mathrm{~km} / \mathrm{h}\) in the north direction and a velocity component of \(15 \mathrm{~km} / \mathrm{h}\) in the eust dircetion. What is the speed of the blimp? What in the dircction of motion of the blimp?

A rock is thrown from a bridge at an upward launch angle of \(30^{\circ}\) with an initial spoed of \(25 \mathrm{~m} / \mathrm{s}\). The bridge is \(30 \mathrm{~m}\) above the river. How moch time clapses before the fock hits the water?
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