a) Average Speed = 1671 meters / 426 seconds ≈ 3.92 m/s
b) The final displacement is approximately 1653.96 meters at an angle of 29.07° south of east.
To solve this problem, we can break it down into two parts: the motion in the first leg and the motion in the second leg. Let's calculate the average speed and final displacement for each part.
First leg (running):
Speed = 4.5 m/s
Time = 3.0 minutes = 3.0 * 60 = 180 seconds
The horizontal component of the velocity (x-component) is given by:
Vx1 = Speed * cos(angle)
Vx1 = 4.5 m/s * cos(25°) ≈ 4.5 * 0.9063 ≈ 4.0934 m/s
The vertical component of the velocity (y-component) is given by:
Vy1 = Speed * sin(angle)
Vy1 = 4.5 m/s * sin(25°) ≈ 4.5 * (-0.4226) ≈ -1.9017 m/s (negative because it's south)
The average speed for the first leg is given by:
Average Speed1 = Total Distance1 / Total Time1
Total Distance1 = Speed * Time
Total Distance1 = 4.5 m/s * 180 s = 810 meters
Average Speed1 = 810 meters / 180 seconds ≈ 4.5 m/s
Second leg (jogging):
Speed = 3.5 m/s
Time = 4.1 minutes = 4.1 * 60 = 246 seconds
The horizontal component of the velocity (x-component) is given by:
Vx2 = Speed * cos(angle)
Vx2 = 3.5 m/s * cos(35°) ≈ 3.5 * 0.8192 ≈ 2.8672 m/s
The vertical component of the velocity (y-component) is given by:
Vy2 = Speed * sin(angle)
Vy2 = 3.5 m/s * sin(35°) ≈ 3.5 * (-0.5736) ≈ -2.0116 m/s (negative because it's south)
The average speed for the second leg is given by:
Average Speed2 = Total Distance2 / Total Time2
Total Distance2 = Speed * Time
Total Distance2 = 3.5 m/s * 246 s = 861 meters
Average Speed2 = 861 meters / 246 seconds ≈ 3.5 m/s
a) Average Speed:
Average Speed = (Total Distance1 + Total Distance2) / (Total Time1 + Total Time2)
Total Distance = Total Distance1 + Total Distance2 = 810 meters + 861 meters = 1671 meters
Total Time = Total Time1 + Total Time2 = 180 seconds + 246 seconds = 426 seconds
Average Speed = 1671 meters / 426 seconds ≈ 3.92 m/s
b) Final Displacement:
To find the final displacement, we need to calculate the x-component and y-component of the displacement for both legs and then add them together.
For the first leg:
Displacement1 (x-component) = Vx1 * Time1 = 4.0934 m/s * 180 s ≈ 736.82 meters
Displacement1 (y-component) = Vy1 * Time1 = -1.9017 m/s * 180 s ≈ -342.31 meters
For the second leg:
Displacement2 (x-component) = Vx2 * Time2 = 2.8672 m/s * 246 s ≈ 705.94 meters
Displacement2 (y-component) = Vy2 * Time2 = -2.0116 m/s * 246 s ≈ -494.83 meters
Total Displacement (x-component) = Displacement1 (x-component) + Displacement2 (x-component) = 736.82 meters + 705.94 meters ≈ 1442.76 meters
Total Displacement (y-component) = Displacement1 (y-component) + Displacement2 (y-component) = -342.31 meters - 494.83 meters ≈ -837.14 meters
To find the magnitude and direction of the final displacement, we can use the Pythagorean theorem and inverse tangent function:
Magnitude of the Final Displacement = sqrt((Total Displacement (x-component))^2 + (Total Displacement (y-component))^2)
Angle of the Final Displacement = atan2(Total Displacement (y-component), Total Displacement (x-component))
Magnitude of the Final Displacement ≈ sqrt((1442.76 meters)^2 + (-837.14 meters)^2) ≈ 1653.96 meters
Angle of the Final Displacement ≈ atan2(-837.14 meters, 1442.76 meters) ≈ -29.07°
Therefore, the final displacement is approximately 1653.96 meters at an angle of 29.07° south of east.