A car moves from a point located 26.0 m from the origin and at an angle of 35.0° from the x-axis to a point located 56.0 m from the origin and at an angle of 60.0° from the x-axis in 2.50 s. What is the magnitude of the average velocity of the car?
Well, the car certainly seems to be going on quite the adventure! To calculate the magnitude of its average velocity, we need to find the displacement of the car and divide it by the time it took.
First, let's break down the initial and final positions of the car into their x and y components.
For the initial position:
x = 26.0 m * cos(35.0°)
y = 26.0 m * sin(35.0°)
And for the final position:
x = 56.0 m * cos(60.0°)
y = 56.0 m * sin(60.0°)
Now, we can use these values to find the displacement in the x and y directions. Let's call them Δx and Δy.
Δx = x_final - x_initial
Δy = y_final - y_initial
Next, we can calculate the magnitude of the displacement using the Pythagorean theorem:
magnitude = √(Δx^2 + Δy^2)
Finally, we divide the magnitude of the displacement by the time to find the magnitude of the average velocity:
average velocity = magnitude / time
So, by plugging in the given values and doing some math, you'll discover the answer!
To find the magnitude of the average velocity of the car, we need to determine the displacement of the car and divide it by the time taken.
Step 1: Calculate the x and y components of the displacement:
The x component is given by: Δx = xf - xi = 56.0 m cos(60.0°) - 26.0 m cos(35.0°)
= 56.0 m * 0.5 - 26.0 m * 0.819
= 28.0 m - 21.3 m
= 6.7 m
The y component is given by: Δy = yf - yi = 56.0 m sin(60.0°) - 26.0 m sin(35.0°)
= 56.0 m * 0.866 - 26.0 m * 0.574
= 48.5 m - 14.9 m
= 33.6 m
Step 2: Calculate the total displacement:
The displacement is given by: Δr = sqrt(Δx^2 + Δy^2)
= sqrt(6.7 m^2 + 33.6 m^2)
= sqrt(44.89 m^2 + 1128.96 m^2)
= sqrt(1173.85 m^2)
= 34.22 m
Step 3: Calculate the magnitude of the average velocity:
The average velocity is given by: v_avg = Δr / t
= 34.22 m / 2.50 s
= 13.69 m/s
Therefore, the magnitude of the average velocity of the car is 13.69 m/s.
To find the magnitude of the average velocity of the car, we need to calculate the displacement and the time taken.
1. Calculate the displacement:
Displacement is the change in position from the initial point to the final point. To find the displacement, we need to subtract the initial position vector from the final position vector.
Let's break down the displacement into its x and y components.
For the initial position (26.0 m, 35.0°), the x-component is given by:
x1 = 26.0 m * cos(35.0°)
And the y-component is given by:
y1 = 26.0 m * sin(35.0°)
For the final position (56.0 m, 60.0°), the x-component is given by:
x2 = 56.0 m * cos(60.0°)
And the y-component is given by:
y2 = 56.0 m * sin(60.0°)
2. Calculate the time taken:
The time taken is given as 2.50 s.
3. Calculate the displacement vector:
The displacement vector is given by the difference between the final and initial position vectors, considering both x and y components.
Displacement vector, Δr = (x2 - x1, y2 - y1)
4. Calculate the magnitude of the average velocity:
The magnitude of the average velocity is given by the formula:
Average Velocity = Displacement / Time Taken
To find the magnitude, we need to calculate the magnitude of the displacement vector, which can be calculated using the Pythagorean theorem:
|Δr| = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Finally, substitute the values into the formula to find the magnitude of the average velocity of the car:
Average Velocity = |Δr| / Time Taken
X = hor. = 26cos35 + 56cos60,
X = 21.3 + 28 = 49.3 m.
Y = ver. = 26sin35 + 56sin60,
Y= 14.9 + 48.5 = 63.4 m.
d^2 = X^2 + Y^2,
d^2 = (49.3)^2 + (63.4)^2,
d^2 = 2430.5 + 4019.6,
d^2 = 6450.1,
d = 80.3 m.
Vavg = d / t = 80.3 / 2.5 = 32.1 m/s.