Calculating Man's Northward Distance Traveled with Vector Components

In summary, the problem involves a man running with a velocity of 5m/s at a direction of 25 degrees north of east for 10 minutes. The question asks for the distance to the north of his starting position after 10 minutes. Using vector components, the distance can be calculated by taking the component of the velocity along the y-axis (North) and using the equation v = x/t, where x is the distance and t is the time. Alternatively, the displacement can be calculated using the equation v = d/t, and then breaking down the displacement vector into components. Both methods should yield the same answer.
  • #1
ch2kb0x
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0

Homework Statement



Man runs with velocity = 5m/s @ 25 degrees north of east for 10 minutes.
How far to the north of his starting position does he end up?

Homework Equations





The Attempt at a Solution


I know this has got to be the easiest problem to solve but I am not getting it. this is what I did.

It asked for how far to the north, so I assumed they were looking for a final position. I used vector components (5cos25)i + 5(sin25)j, but I don't know how to solve for the distance x. I know there's a formula v = x / t but that didnt help.
 
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  • #2


Vector components is the way to go. They only ask for the distance traveled North, so use the component of the velocity along the y-axis (North) and from that magnitude (speed), you can use your Speed = distance/time equation.

i.e. x = vt

(remember to convert time to seconds).

OR

You could work out his displacement, using Velocity = displacement/time, i.e. Displacement = velocity x time and then break the displacement vector down into components.

Both should give you the same answer.
 
  • #3


I would approach this problem by first defining the variables and setting up a coordinate system. In this case, the man's velocity can be represented by the vector v = 5m/s @ 25 degrees north of east. We can define the x-axis as east and the y-axis as north.

Next, we can use the formula v = x/t to solve for the distance traveled in the x-direction (east). We know that the velocity in the x-direction is given by v_x = 5m/s cos(25) = 4.55m/s. We also know that the time of travel is 10 minutes or 600 seconds. Therefore, we can calculate the distance traveled in the x-direction as x = (4.55m/s)(600s) = 2730m.

To calculate the distance traveled in the y-direction (north), we can use the same formula v = y/t. The velocity in the y-direction is given by v_y = 5m/s sin(25) = 2.08m/s. Therefore, the distance traveled in the y-direction is y = (2.08m/s)(600s) = 1248m.

To find the total distance traveled in the northward direction, we can use the Pythagorean theorem. The total distance d is given by d = √(x^2 + y^2) = √(2730m^2 + 1248m^2) = √(3972964m^2) = 1993m.

Therefore, the man ends up 1993m north of his starting position.
 

1. What are vector components?

Vector components are the individual parts of a vector that represent its direction and magnitude. They are typically represented as x and y components in a two-dimensional space, or x, y, and z components in a three-dimensional space.

2. How do vector components help in scientific research?

Vector components are essential in various scientific fields, such as physics, engineering, and mathematics. They help in analyzing and understanding the motion, forces, and other properties of objects in a given system. They also aid in solving complex problems and making accurate predictions.

3. How are vector components calculated?

Vector components can be calculated using trigonometric functions such as sine, cosine, and tangent. The magnitude of a vector can be determined using the Pythagorean theorem, while its direction can be found using inverse trigonometric functions.

4. What is the difference between scalar and vector components?

Scalar components have only magnitude, while vector components have both magnitude and direction. Scalar components can be added using simple arithmetic, while vector components require vector addition, which takes into account their direction as well as magnitude.

5. Can vector components have negative values?

Yes, vector components can have negative values. Negative components indicate that the vector is pointing in the opposite direction to the positive component. This allows for a more accurate representation of the vector's direction in a given system.

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