A truck is moving north at a speed of 70 km/h. The exhause pipe above the truck cab seeds out a tail of smoke that makes an angle of 20 degree east of south behind the truck. If the wind is flowing directly towards the east. What is the speed at that location?
Solutions. Showing all the working
To find the speed at the location where the tail of smoke is making an angle east of south behind the truck, we need to analyze the vectors involved:
1. The truck is moving north at a speed of 70 km/h. This can be represented as a vector pointing directly north.
2. The wind is flowing directly towards the east. This can be represented as a vector pointing directly east.
3. The tail of smoke forms an angle of 20 degrees east of south behind the truck. To represent this vector, we need to break it down into its east and south components.
Now, we can use vector addition to find the resulting velocity at that location. To do this, we need to find the resultant vector by adding the velocities of the truck and the wind.
Here's how we can calculate it step by step:
1. Convert the speed and direction of the truck into vector form:
- The velocity vector of the truck (V1) would be 70 km/h towards the north.
2. Convert the wind speed and direction into vector form:
- The wind speed vector (V2) would be the velocity of the wind, which is directly towards the east.
3. Convert the smoke tail direction into vector form:
- The smoke tail angle is 20 degrees east of south. We can break down this vector into its east and south components by using trigonometry.
- The east component (V3_east) can be calculated by multiplying the magnitude of the smoke tail velocity (which is unknown) by the cosine of the angle.
- The south component (V3_south) can be calculated by multiplying the magnitude of the smoke tail velocity by the sine of the angle.
4. Sum the vectors:
- The resultant vector (V_resultant) can be calculated by adding the vectors V1, V2, V3_east, and V3_south together.
5. Calculate the magnitude of the resultant velocity:
- The magnitude of the resultant velocity will give us the speed at that location.
Although we don't have the exact values of the magnitudes or distances, you can substitute the values you have into the equations to find the resultant velocity magnitude, which represents the speed at the location where the tail of smoke is making an angle east of south behind the truck.
70 sin20° = 23.9 km/hr
Think about it. If there were no wind, the smoke would go straight back, appearing to go south
If the wind were blowing at 70 km/hr east, the smoke would go as far east as it did south, making a 45° angle.
So, the more it appears to go to the south, the slower the wind is blowing to the east.
i dont think your is correct because if you assume the angle is 45 in this case the you do 70sin45 it does not give 70km/h
i think the more correct one is using a tan because 70tan45 gives you 70km/h
Well, it looks like the truck is going north and the wind is going east. That sounds like a typical family road trip where dad refuses to ask for directions and mom just wants to stop for snacks.
So, with the wind blowing east, it won't really affect the speed of the smoke coming out of the exhaust pipe. It's just going to make it blow in a diagonal direction.
Since the exhaust pipe is spewing smoke in a direction 20 degrees east of south, we can imagine a right-angled triangle formed with the sides being the northward speed of the truck and the speed of the smoke, and the hypotenuse being the resultant speed of the smoke.
Using a little trigonometry magic and some dad jokes, we can find that the resultant speed of the smoke is approximately 73.7 km/h. So, if you're ever in that location, make sure to bring some extra tissues for the sneezes caused by the smoke.