Speed, Displacement, and Velocity Question

In summary, the boat must be pointed at an angle of 37 degrees upstream and it takes 1.5x10^2 seconds for the boat to cross the river. The Pythagorean theorem and the use of vector components are necessary to solve this problem.
  • #1
Jordan_
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Homework Statement



A boat, propelled so as to travel with a speed of 0.50m/s in still water, moves directly across a river that is 60m wide. The river flows with a speed of 0.30m/s. (a) At what angle, relative to the straight across direction, must the boat be pointed? (b) How long does it take the boat to cross the river?

Answer: (a) 37 degrees upstream (b) 1.5x10^2 seconds.

Homework Equations



pythagorean theorem


The Attempt at a Solution



I tried to use the movement of the boat at 0.50m/s with the 60m across and found that that would take 120 seconds. Then, I multiplied the 120 seconds by the river current speed of 0.30m/s to find that it would pull the boat 36 meters downstream. Then I tried to find the angle using tan(theta) = opp/adj but I got the angle 31 degrees.

I tried several other methods last night that I can't remember now, and I kept getting 31 degrees. The book states that it is 37 degrees, however, and so I'm wondering what I'm doing wrong.
 
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  • #2
I tried to use the movement of the boat at 0.50m/s with the 60m across and found that that would take 120 seconds.

This isn't correct because the boat's velocity will have two components to it since it must head at an angle due to the river current. You will need to figure out the component that is perpendicular to the river for part (b).

So don't worry about using the distance just yet. Draw your vectors first. You know the direction and speed of the river, and you know the speed the boat can go relative to the still water. So if the boat is to land directly across from its starting point, you know the boat must be initially pointing upstream at some angle. Adding these two vectors must give a vector that goes straight across the river. So draw a triangle based on this, and you can solve for the angle.
 
  • #3


It seems like you are on the right track with using the Pythagorean theorem and the trigonometric function to find the angle. However, it is important to keep in mind that the boat's velocity is a combination of its speed and direction, which is affected by the river's current.

To find the correct angle, you can use the following equation: tan(theta) = (0.30m/s)/(0.50m/s). This gives you a theta value of approximately 37 degrees, which is the correct answer according to the book.

Additionally, to find the time it takes for the boat to cross the river, you can use the formula time = distance/speed. In this case, the distance is 60m and the speed is 0.50m/s. This gives you a time of 120 seconds, which is the correct answer according to the book.

Overall, it is important to carefully consider all the factors and how they affect the boat's movement in order to find the correct solutions for displacement, velocity, and time. Keep up the good work!
 

What is the difference between speed and velocity?

Speed is a measure of how fast an object is moving, while velocity is a measure of both the speed and direction of an object's motion. In other words, velocity includes information about the direction an object is moving, while speed does not.

How is displacement different from distance?

Distance is a measure of how far an object has traveled, while displacement is a measure of the straight-line distance and direction from an object's starting point to its ending point. Distance is a scalar quantity, while displacement is a vector quantity.

What is the formula for calculating speed?

The formula for calculating speed is speed = distance / time. This means that speed is equal to the distance traveled divided by the time it took to travel that distance.

How is average speed different from instantaneous speed?

Average speed is a measure of the total distance an object travels divided by the total time it takes to travel that distance. Instantaneous speed, on the other hand, is the speed of an object at a specific moment in time. It is calculated by finding the slope of the distance-time graph at that point.

Can an object have a negative displacement or velocity?

Yes, an object can have a negative displacement or velocity. This means that the object is moving in the opposite direction from its starting point or has a negative change in position. For example, an object moving westward would have a negative displacement if its starting point is in the east.

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