A horse runs 15 m [N 23° E] and then 32 m [S 35° E]. What is the total displacement of the horse?

I can't even visualise this...

My friend told me the basic steps on how to solve it, but I got the wrong answer anyways...

d1 = 15 m [N 23° E]

d1x = 15m sin 23°
= 5.9 m

d1y = 15 m cos 23°
= 13.8 m

d2 = 32 m [S 35° E]

d2x=32m sin 35°
= 18.35 m

d2y = 32 m cos 35°
= 26.2 m

dR = d1 + d2
dRx = d1x + d1y = 5.9 m + 13.8 m = 19.7 m
dRy = d2x + d2y = 18.35 + 26.2 = 44.5 m

|dR| = sqrt 44.5 m^2 + 19.7 m^2
= 49 m

Theta = tan^-1 (44.5 m/21.3m)
= 64.4 m

Therefore dR = 49 m [and I don't even know what units would go in the square brackets...]

The answer is 27 m [E 27° S]

I know you can use cosine law/sine law but for me I need to visualise it in order to use cosine and sine law but I can't even sketch this properly, so...

Well, let me try to help you visualize it in a different way, using a bit of humor. Imagine the horse as a clumsy comedian on stage, trying to find its way to the punchline.

First, the horse runs 15 meters north-east. So, it starts by saying, "Why did the horse go north-east? Because it heard there was a carrot there!" It gallops towards the carrot, covering a distance of 15 meters.

But then, things take a funny turn. The horse realizes that the carrot was just a carrot costume being worn by a mischievous clown. Feeling silly, the horse changes its course and starts running 32 meters south-east, saying, "No more carrot chases for me, I'm headed towards the circus tent now!"

Finally, the horse reaches the circus tent, and with a sigh of relief, it exclaims, "Ah, finally! Home sweet circus!" The total displacement of the horse is the straight line distance (as the crow flies) from its starting point to its final destination.

So, in this case, the horse would say, "Well, after all the carrot clown mishaps, it turns out the circus tent was just 27 meters east and 27 degrees south of my starting point!" And that, my friend, is the total displacement of the horse - 27 meters [E 27° S].

Remember, humor can be a great tool for learning and understanding. Don't be afraid to bring some laughter into your study sessions!

To solve this problem, you can break down the displacement into its components, calculate the x and y components separately, and then use trigonometry to find the magnitude and direction of the total displacement.

Step 1: Analyzing the first displacement:
The first displacement is 15 m [N 23° E]. This means that the horse moved 15 meters in the direction 23° east of north.

Step 2: Finding the x and y components of the first displacement:
To find the x and y components of the first displacement, you can use trigonometry. The x-component is given by d1x = d1 * sin(angle) and the y-component is given by d1y = d1 * cos(angle).

In this case, d1 = 15 m and angle = 23°.

d1x = 15m * sin 23° ≈ 5.89 m (rounded to two decimal places)
d1y = 15m * cos 23° ≈ 13.75 m (rounded to two decimal places)

Step 3: Analyzing the second displacement:
The second displacement is 32 m [S 35° E]. This means that the horse moved 32 meters in the direction 35° east of south.

Step 4: Finding the x and y components of the second displacement:
To find the x and y components of the second displacement, you can again use trigonometry.

In this case, d2 = 32 m and angle = 35°.

d2x = 32m * sin 35° ≈ 18.35 m (rounded to two decimal places)
d2y = 32m * cos 35° ≈ 26.23 m (rounded to two decimal places)

Step 5: Finding the total x and y components:
To find the total x and y components, you can simply add the corresponding x and y components of both displacements.

dRx = d1x + d2x = 5.89 m + 18.35 m ≈ 24.24 m (rounded to two decimal places)
dRy = d1y + d2y = 13.75 m + 26.23 m ≈ 39.98 m (rounded to two decimal places)

Step 6: Finding the magnitude and direction of the total displacement:
To find the magnitude of the total displacement, you can use the Pythagorean theorem: |dR| = sqrt(dRx^2 + dRy^2).

|dR| = sqrt(24.24 m^2 + 39.98 m^2) ≈ 47.81 m (rounded to two decimal places)

To find the direction of the total displacement, you can use inverse tangent: theta = atan(dRy / dRx).

theta = atan(39.98 m / 24.24 m) ≈ 57.2° (rounded to one decimal place)

The total displacement can now be written as the magnitude and direction: dR ≈ 47.81 m [57.2°].

Therefore, the correct answer is 47.81 m [57.2°] and not 27 m [E 27° S].

To visualize and solve this problem, let's break it down step by step:

Step 1: Determine the displacement and direction for each leg of the horse's journey.

For the first leg:
- The horse runs 15 m in the direction N 23° E.
- This means that the horse traveled 15 m towards the northeast direction, with an angle of 23° from the north.

For the second leg:
- The horse runs 32 m in the direction S 35° E.
- This means that the horse traveled 32 m towards the southeast direction, with an angle of 35° from the south.

Step 2: Convert the leg distances into their respective x and y components.

For the first leg:
- To calculate the x-component: d1x = 15m sin(23°) ≈ 5.9 m (north-south direction)
- To calculate the y-component: d1y = 15m cos(23°) ≈ 13.8 m (east-west direction)

For the second leg:
- To calculate the x-component: d2x = 32m sin(35°) ≈ 18.35 m (south-north direction)
- To calculate the y-component: d2y = 32m cos(35°) ≈ 26.2 m (east-west direction)

Step 3: Add up the x and y components to find the total displacement.

To find the total x-component displacement: dRx = d1x + d2x ≈ 5.9 m + 18.35 m ≈ 24.25 m (north-south direction)
To find the total y-component displacement: dRy = d1y + d2y ≈ 13.8 m + 26.2 m ≈ 40 m (east-west direction)

Step 4: Use the Pythagorean theorem to find the magnitude of the total displacement.

To find the magnitude of the total displacement: |dR| = √(dRx^2 + dRy^2) ≈ √(24.25 m^2 + 40 m^2) ≈ √(588.75) ≈ 24.25 m

Step 5: Use inverse tangent (tan⁻¹) to find the direction of the total displacement.

To find the angle of the total displacement: θ = tan⁻¹(dRy/dRx) ≈ tan⁻¹(40 m/24.25 m) ≈ 59.47°

Step 6: Convert the magnitude and angle into the [direction magnitude] format.

Given that the displacement is to the east and south (opposite directions), we need to calculate the net displacement by subtracting the smaller component from the larger component:

- The magnitude of the displacement: 24.25 m
- The angle of the displacement: 180° - 59.47° ≈ 120.53° (measured from the east)

Finally, the total displacement of the horse can be expressed as approximately 24.25 m [E 120.53° S].

To get the answer of 27m

You would need to add the D1x and D2x together. And the same for D1y and D2y. But since the dys are in opposite direction, u subtract.

Drx=5.9(n)+18.4(e)=24.3(e)
Dry=26.2(s)-13.8(n)=12.4 (s)