An ant walks on a piece of graph paper straight along the 𝓍 a...

Question

An ant walks on a piece of graph paper straight along the 𝓍 axis a distance of 10.0 cm in 2.40 s. It then turns left 40.0° and walks in a straight line another 10.0 cm in 1.80 s. Finally, it turns another 70.0° to the left and walks another 10.0 cm in 1.55 s. Determine its magnitude and direction.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A small robot navigates a grid on the laboratory floor. Initially moving 12.0 centimeters straight across the X axis in 2.50 seconds. It then turns 45 degrees to the left and continues straight for another 12.0 centimeters in 2.00 seconds. Finally, it turns an additional 55 degrees to the left, moving another 12.0 centimeters in 1.60 seconds, calculate its magnitude and direction. So that's our end goal. Our final answer that we're ultimately trying to solve for is we're trying to figure out what the magnitude and direction of the robot is. Awesome. We're also given some multiple choice answers. All of our units for the magnitude are in centimeters per second and all the direction units are in degrees. So let's read them off to see what our final answer pair might be. A is 3.41 and 41.3 B is 4.50 and 47.8 C is 5.55 and 54.0. And D is 6.05 and 62.0. Awesome. So first off, let us create a diagram to better visualize this problem. So I've gone ahead and put together a fancy diagram here. So on our far left of our diagram, we have the Y axis and we have 0.0 or 0.0 I should say represents the origin point. And the small robot will navigate on a grid on the lavatory floor, initially moving 12 centimeters in a straight line across the X axis. So from O to point E going 12.0 centimeters in 2.50 seconds and then it will turn 45 degrees to the left and continue straight for another 12.0 centimeters in 2.00 seconds. And then it'll go from point E to point F and then from point F, it will turn 55 degrees to the left and move another 12.0 centimeters in 1.60 seconds to head to point G. And then finally, it will turn another 55 degrees to, I'm sorry. That's so then I'll go to point G Awesome. So from point G, so from point A to point G, that will be our magnitude and direction will be that vector is what we're trying to investigate and solve for OK. So moving right along here. Now, if I hat and J hat are unit vectors found along the X and Y axis, respectfully, we can therefore, therefore, we can describe the robot's journey across the laboratory grid as follows. So we could say that vector oe is equal to 12.0 centimeters I hat. And we can also say that vector ef is equal to 12.0 multiplied by oh, don't forget, centimeters. So it's 12.0 centimeters. We don't wanna forget our units. 12.0 centimeters multiplied by cosine of 45.0 degrees. I had plus 12.0 centimeters multiplied by sign of 45.0 degrees. J hat in its units are centimeters which we already did that great. So let us note that vector FG. So now we're focusing on FG vector FG, we'll make an angle at. So this is angle at 45.0 degrees plus 55.0 degrees, which is equal to 100.0 degrees with the positive X axis. Thus, we can write that vector FG is equal to 12.0 multiplied by cosine of 100.0 degrees. I had plus 12.0 multiplied by sine of 100.0 degrees J hat centimeters. Awesome. So let us also recall that the average velocity is equal to the total displacement divided by the total time. And using this equation, we could write that vector V average is equal to 12.0 centimeters. I hat plus 12.0 multiplied by cosine of 45.0 degrees. I hat plus 12.0 multiplied by sine of 45.0 degrees. J hat centimeters plus 12.0 multiplied by cosine of 100.0 degrees. I hat plus 12.0 multiplied by sine of 100.0 degrees. J hat centimeters all divided by 2.50 seconds plus 2.00 seconds plus 1.60 seconds. Awesome. Which we can simplify. To vector V average is equal to 18.40 I hat plus 20.30 J hat centimeters divided by 6.10 seconds. Awesome, which we can simplify and solve for the V average is, or I should say vector V average is equal to 3.02 I hat plus 3.33 J hat and its units are centimeters awesome. So from this information, we can determine that for part I, the X component of the robot's average velocity. So V so the vector V average X will equal 3.02 centimeters per second. And for part I I, the Y component of the ro robots average velocity. So vector V average Y will equal 3.33 centimeters per second. And finally, for part II I, the magnitude of the robot's average velocity which will equal so vector V average. So V average is equal to the square root of 3.02 centimeters per second squared. So we don't get confused. That's a three plus because we have to take the magnitude. So we take the square root of the I hat value plus the J hat value and each of those values are squared. So it's I hat squared plus J hat squared. So 3.02 centimeters per second squared plus 3.33 centimeters per second squared, which is equal to 4.50 when we round to two decimal places centimeters per second. And that is our final answer. For part I I or we, I mean II I, so our third answer. So we have our first part answer. So part I I is equal to 3.02 centimeters per second. And for I I, it is equal to 3.33 centimeters per second. And then finally, for II I, our third answer is equal to 4.50 centimeters per second. Awesome. And now for our final answer, which is the direction of the robots average velocity relative to the positive X axis is written as theta is equal to tangent or inverse tangent I should say. So theta is equal to inverse tangent of 3.33 centimeters per second, divided by 3.02 centimeters per second, our units will cancel out. And when we plug that into our calculator, we will get 47.8 degrees when we round to one decimal place. And this is measured from the counter clockwise direction. And that is our final answer for the direction. So we did it, we found our final answer answers, I should say direction. So our direction is 47.8 degrees and our magnitude, which is the answer we needed to solve for is 4.50 centimeters per second. Awesome. So going back to the top here to look at our multiple choice answers, the correct answer has to be the letter B 4.50 centimeters per 2nd and 47.8 degrees. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Vector Addition

Magnitude and Direction

Trigonometry in Physics

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