A train at a constant 44.0 km/h moves east for 42 min, then in a direction 45.0° east of due north for 17.0 min, and then west for 51.0 min. What are the (a) magnitude (in km/h) and (b) angle (relative to north, with east of north positive and west of north negative) of its average velocity during this trip?
s1=v •t1=44•42/60 =30.8 km
s2 =v•t2 = 44•17/60 = 12.5 km
s3 = v•t3 = 44•51/60=37.4 km
Cosine Law:
displacement d= sqrt[(s2² +(s3-s1)²-2•s2•(s3-s1) •cos135] = 17.8 km
ave velocity = d/time taken = 17.8• 60/(42+17+51) =9.7 m/s
Sine Law:
(s3-s1)/sinα =d/sin135
sinα = sin135(s3-s1)/d = 0.707•6.6/17.8 =0.26.
α =15.07°
β=135-90=45°
θ= α+ β =15.07°+45° =60.07°
To find the average velocity, we need to calculate the displacement of the train and the total time taken for the trip.
Let's break down the train's motion into three segments:
Segment 1: Moving east for 42 min at a constant speed of 44.0 km/h.
Segment 2: Moving in a direction 45.0° east of due north for 17.0 min.
Segment 3: Moving west for 51.0 min.
To find the displacement for each segment, we can use the formula:
Displacement = Velocity * Time
Segment 1:
Velocity = 44.0 km/h
Time = 42 min = 0.7 hours
Displacement1 = 44.0 km/h * 0.7 hours
Segment 2:
Velocity = 44.0 km/h (constant velocity on the North direction)
Time = 17 min = 0.28 hours
Displacement2 = 44.0 km/h * 0.28 hours
Segment 3:
Velocity = -44.0 km/h (moving in the opposite direction to its initial velocity)
Time = 51 min = 0.85 hours
Displacement3 = -44.0 km/h * 0.85 hours
To find the total displacement, we can sum up the individual displacements:
Total Displacement = Displacement1 + Displacement2 + Displacement3
Now, let's calculate the average velocity:
Total Time = Time for Segment 1 + Time for Segment 2 + Time for Segment 3
Total Time = 0.7 hours + 0.28 hours + 0.85 hours
Now we have all the values needed to find the average velocity:
(a) Average Velocity (magnitude) = Total Displacement / Total Time
(b) Average Velocity (angle) = Angle of the Total Displacement with respect to the North direction
To find the angle, we can use trigonometry:
Angle = arctan(Vertical Component of Displacement / Horizontal Component of Displacement)
Now let's calculate the values:
Substituting the given values:
Displacement1 = 44.0 km/h * 0.7 hours = 30.8 km
Displacement2 = 44.0 km/h * 0.28 hours = 12.32 km
Displacement3 = -44.0 km/h * 0.85 hours = -37.4 km
Total Displacement = 30.8 km + 12.32 km + (-37.4 km) = 5.72 km
Total Time = 0.7 hours + 0.28 hours + 0.85 hours = 1.83 hours
(a) Average Velocity (magnitude) = Total Displacement / Total Time = 5.72 km / 1.83 hours ≈ 3.13 km/h
The magnitude of the average velocity is approximately 3.13 km/h.
To find the angle, we need to calculate the components of the total displacement:
Vertical Component of Displacement = Displacement2 = 12.32 km
Horizontal Component of Displacement = Displacement1 + Displacement3 = 30.8 km + (-37.4 km) = -6.6 km
Angle = arctan(12.32 km / -6.6 km) ≈ -60.2°
The angle of the average velocity, relative to north, with east of north positive and west of north negative, is approximately -60.2°.