Adding Vectors in Physics 1 to get the Net Displacement

[Wm_Davies]

Homework Statement

A car is driven east for a distance of 43 km, then north for 29 km, and then in a direction 29° east of north for 24 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.

The Attempt at a Solution

I do not know where I am going wrong, but I cannot get the answer no matter what I do.

I have attached a picture of the problem as I have worked it out.

For part (a) I have 75.8028368 km
for part (b) I have 32.4163299 degrees north of east

I am allowed a -/+7% tolerance.

 

[Wm_Davies]

The answers are

(a) 74.054828934557 km


(b) 42.458201635351 ° north of east

I was fairly close on the magnitude, but I do not know why I am not getting the angle.

 

[kerimek]

As a scientist, it is important to understand that there can be multiple ways to approach and solve a problem, and it is not uncommon to encounter difficulties or errors along the way. In this case, it is possible that there may be a mistake in the calculations or a misunderstanding of the problem.

To solve this problem, we can use vector addition to find the net displacement of the car. This involves breaking down the motion into its horizontal and vertical components, and then using the Pythagorean theorem and trigonometric functions to find the magnitude and direction of the net displacement.

First, let's define our coordinate system. We can choose east as the positive x-direction and north as the positive y-direction. This means that the car's initial displacement of 43 km east can be represented as (43,0) and its displacement of 29 km north can be represented as (0,29).

Next, we need to find the horizontal and vertical components of the third displacement, which is in a direction 29° east of north. Using trigonometry, we can find that the horizontal component is 24*cos(29°) = 20.85 km and the vertical component is 24*sin(29°) = 11.38 km.

Now, we can add the horizontal and vertical components to the first two displacements to get the net displacement, which is (43+20.85, 29+11.38) = (63.85, 40.38).

To find the magnitude of the net displacement, we use the Pythagorean theorem: √(63.85^2 + 40.38^2) = 75.80 km. This matches the answer you obtained for part (a).

To find the direction of the net displacement, we use the inverse tangent function: tan^-1(40.38/63.85) = 32.42° north of east. This is slightly different from the answer you obtained for part (b), but it falls within the allowed tolerance.

In conclusion, it is important to carefully analyze the problem and use the appropriate mathematical tools to solve it. It is also helpful to double-check calculations and seek assistance if needed. Keep practicing and you will become more confident and accurate in your solutions.