Vector Question?

Question

I have no idea how to do this problem. I have the answer though, it is 49m at 7.3° to the right of down field

Here is the problem:

A football player runs directly down the field for 35m before turning to the right at an angle of 25° from his original direction and running an additional 15m before getting tackled. What is the magnitude and direction of the runner's total displacement?

Answer 1

Easy. Draw triangles.

First, draw a line 35m downfield. Now draw a line at an angle of 25 degrees to the right for 15m.

Now extend the first (35m) line indefinitely in its original direction. Now draw a line from the far end of the 25-degree line perpendicular to the first line. You now have one little right triangle with a hypotenuse of 15m and an angle of 25 degrees.

Figure out how long the extension is from the point of turning to the perpendicular: 15 cos 25.

The extended line in the original direction thus has length

35 + 15 cos 25

The prependicular piece has length

15 sin 25

Now draw one long line from the original starting point to the ending point. You have a large right triangle with sides

15 sin 25
and
35 + 15 cos 25

You can figure out the hypotenuse and angles of that big triangle.

Answer 2

Start by drawing a line 35 units long on a page, left to right. Lets use centimeters for simplicity. Put an arrowhead on the right end of the line. Then, using a protractor, determine where 25 degrees will take you from the arrow. Starting at the arrow, draw a line 15 cm long in the direction you discovered with the protractor. Put an arrow at the end of the new line. Now, draw a third line that joins the beginning of the first line with the end of the second line. This represents the sum of the two vectors, otherwise called the displacement. Measure the third line and you'll have the actual distance travelled (as the crow flies). Determine with your protractor the angle between the first line and the third line and you have the direction.

Of course, you don't need to measure anything to figure out the problem... you can do it all with basic trigonometry. If you already knew that, though, I doubt you'd be asking the question.

Answer 3

The total distance down field will be 35 + 15*cos(25)=48.59
The total distance ran to the right is: 15*sin(25)=6.33
Next, to find the magnitude, use the pythagorian theorem: sqrt((48.59)^2+(6.33)^2)=49
To calculate the angle, compute arctan(6.33/48.59)=7.4 degrees to the right (a little different from yours but I double checked and my calculation is right).

Answer 4

let y = downfield and x = to the right
y = 35 + 15cos 25, x = 15sin25
d^2 = x^2 +y^2 @ arctan(x/y) or d = y/cos(arctan(x/y))
y = 48.5946, x = 6.33927
φ =7.43288°
d = 49.00636