A football player runs for a distance d1 = 8.57 m in 1.21 s, at an angle of θ = 62.9 degrees to the 50-yard line, then turns left and runs a distance d2 = 10.58 m in 1.59 s, in a direction perpendicular to the 50-yard line. The diagram shows these two displacements relative to an x-y coordinate system, where the x axis is parallel to the 50-yard line, and the y axis is perpendicular to the 50-yard line.
What is the magnitude of the average velocity, in m/s?
What angle, in degrees does the average velocity make with the y axis? (Note that the angle θ was given as measured from the x axis rather than the y axis.)
i dont know
Well, this football player sure knows how to make things interesting! Let's see if we can tackle these questions about their average velocity with some humor.
To find the magnitude of the average velocity, we'll need to calculate the total displacement of the football player and divide it by the total time taken.
Now, let's sum up the distances first: d1 + d2 = 8.57 m + 10.58 m = 19.15 m. To find the total time taken, we add the individual times: 1.21 s + 1.59 s = 2.8 s.
So, the football player traveled a total distance of 19.15 m in 2.8 seconds. Now, we divide the total distance by the total time: 19.15 m / 2.8 s ≈ 6.84 m/s.
Therefore, the magnitude of the average velocity is approximately 6.84 m/s.
Now, let's move on to the angle that the average velocity makes with the y-axis. Since the angle θ was given as measured from the x-axis, we need to subtract it from 90 degrees to get the angle with the y-axis.
So, 90 degrees - 62.9 degrees = 27.1 degrees.
Therefore, the average velocity makes an angle of approximately 27.1 degrees with the y-axis.
Remember, these calculations are just for fun. Don't take them too seriously, just like a clown on a unicycle.
To find the magnitude of the average velocity, we can use the formula:
Average velocity = total displacement / total time
First, let's calculate the total displacement by adding the two displacements together. Since the second displacement is perpendicular to the 50-yard line, it only contributes to the y component of the displacement.
d1 = 8.57 m
d2 = 10.58 m
Total displacement (Δd) = √((d1)^2 + (d2)^2)
= √((8.57)^2 + (10.58)^2)
= √(73.6649 + 111.7764)
= √185.4413
= 13.60 m (rounded to two decimal places)
Next, let's calculate the total time:
t1 = 1.21 s
t2 = 1.59 s
Total time (Δt) = t1 + t2
= 1.21 s + 1.59 s
= 2.80 s (rounded to two decimal places)
Now, let's calculate the average velocity:
Average velocity = Δd / Δt
= 13.60 m / 2.80 s
= 4.86 m/s (rounded to two decimal places)
The magnitude of the average velocity is 4.86 m/s.
To find the angle the average velocity makes with the y-axis, we can use the inverse tangent function:
θ = atan(d1 / d2)
= atan(8.57 m / 10.58 m)
= atan(0.8102)
= 40.67 degrees (rounded to two decimal places)
Note that this angle is given with respect to the x-axis. To find the angle with respect to the y-axis, we need to subtract it from 90 degrees:
Angle with respect to y-axis = 90 degrees - 40.67 degrees
= 49.33 degrees (rounded to two decimal places)
Therefore, the angle the average velocity makes with the y-axis is 49.33 degrees.
To find the magnitude of the average velocity, we need to calculate the total displacement of the football player and divide it by the total time elapsed.
First, let's calculate the total displacement. Since the player moves at an angle of 62.9 degrees to the 50-yard line, we can use trigonometry to find the x and y components of this displacement.
The x component, dx, can be found using the formula:
dx = d1 * cos(θ)
dx = 8.57 m * cos(62.9°)
Similarly, the y component, dy, can be found using the formula:
dy = d1 * sin(θ)
dy = 8.57 m * sin(62.9°)
Next, let's calculate the displacement for the second part of the motion. Since the player runs in a direction perpendicular to the 50-yard line, the displacement only has a y component.
dy2 = d2
Now, we can find the total displacement by adding the x and y components:
Displacement = √(dx + dy)^2
Displacement = √[(dx + 0)^2 + (dy + dy2)^2]
To find the average velocity, we divide the displacement by the total time:
Average velocity = Displacement / total time
Average velocity = √[(dx + 0)^2 + (dy + dy2)^2] / (1.21 s + 1.59 s)
After calculating the above expression, we get the magnitude of the average velocity.
To find the angle that the average velocity makes with the y-axis, we can use the inverse tangent function:
Angle = arctan(dx / (dy + dy2))
After calculating this expression, we get the angle in degrees.
By following these calculations, you can find the magnitude of the average velocity and the angle it makes with the y-axis.