Riding the Train -Vecotrs & components

In summary, a passenger on a northbound train traveling at 16.0 mi/hr walks eastward at 5.4 mi/hr. Using Pythagoras' Theorem, the resultant speed of the passenger with respect to the ground is approximately 16.9 mi/hr. However, the numerical direction of the passenger cannot be determined without knowing the angle at which they are walking.
  • #1
junesmrithi
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0
Riding the Train --Vecotrs & components

A passenger train is cruising directly to the north at 16.0 mi/hr.
what are the speed and (numerical) direction of a passenger with respect to the ground on this train, if he now walks east-ward at 5.4 mi/hr?
---Compass heading in degrees: N=0, E=90, S=180, and W=270

I know i have to add velocities add as vectors.

Work-
i drew a triagle, the y commponent being 16 and the x (hoizontal) component being 5.4 and i ahve to ifnd the resultant speed. aka: the hypothenuse of the triangle.
But i don't know how to do that when i don't have an angle.
Please help
 
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  • #2
Hint: Pythagoras' Theorem.
 
  • #3


I would like to clarify a few things about vectors and components in this situation. Firstly, velocity is a vector quantity, which means it has both magnitude (speed) and direction. In this case, the train's velocity is 16.0 mi/hr due north, which can be represented as a vector pointing upwards on a graph.

Now, when the passenger starts walking eastward at 5.4 mi/hr, we can also represent this as a vector pointing to the right on the same graph. This is known as the passenger's velocity relative to the train.

To find the passenger's velocity with respect to the ground, we need to find the resultant vector by adding the two velocity vectors together. This can be done using vector addition, which involves finding the sum of the x-components and y-components separately.

In this case, the x-component of the train's velocity is 0 mi/hr (since it is moving directly north) and the x-component of the passenger's velocity is 5.4 mi/hr. So, the resultant x-component is 0 + 5.4 = 5.4 mi/hr.

Similarly, the y-component of the train's velocity is 16.0 mi/hr and the y-component of the passenger's velocity is 0 mi/hr (since the passenger is not moving vertically). So, the resultant y-component is 16.0 + 0 = 16.0 mi/hr.

Now, we can use the Pythagorean theorem to find the magnitude (or speed) of the resultant vector, which is the hypotenuse of the triangle you drew. The formula is c = √(a^2 + b^2), where c is the hypotenuse (resultant speed), a is the x-component (5.4 mi/hr), and b is the y-component (16.0 mi/hr).

Plugging in the values, we get:
c = √(5.4^2 + 16.0^2) = √(29.16 + 256) = √285.16 = 16.88 mi/hr

So, the passenger's speed with respect to the ground is 16.88 mi/hr. To find the direction, we can use trigonometric functions to find the angle between the resultant vector and the horizontal axis.

The tangent of this angle is the opposite side (16.0 mi/hr) divided by the adjacent side (
 

FAQ: Riding the Train -Vecotrs & components

What is the difference between a vector and a component?

A vector is a quantity that has both magnitude and direction, while a component is one part of a vector that acts in a specific direction. Vectors can be broken down into two or more components.

How do you calculate the magnitude of a vector?

The magnitude of a vector is the length of the vector, and it can be calculated using the Pythagorean theorem. If the vector has components in the x and y directions, the magnitude can be found using the formula: magnitude = sqrt(x^2 + y^2).

What are the standard units for measuring vectors?

Vectors are typically measured in units of length, such as meters or feet. However, they can also be measured in other units depending on the context, such as force (Newton) or velocity (meters per second).

How do you find the direction of a vector?

The direction of a vector is typically given by an angle measured counterclockwise from a reference direction, such as the positive x-axis. This angle can be found using trigonometric functions, such as tangent or sine.

Can a vector have more than two components?

Yes, a vector can have any number of components depending on the situation. For example, in a three-dimensional space, a vector can have components in the x, y, and z directions. In physics, vectors can also have components in multiple dimensions, such as time.

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