A car’s initial velocity is 50.0 km/h in the direction 60.0° north of east, and its final velocity is 70.0 km/h in the direction 40.0° south of east. If the time period for this journey is 30.0 minutes, what is the magnitude of the car’s average acceleration?
a=(vf-vi)/time
There are a number of ways to change the vectors, I recommend you break each up into N, and E components, then solve acceleration in N, and acceleration in E, then add them.
the magnitude of average acceleration will be sqrt (anorth^2 + aeast^2)
I DONT UNDERSTAND D QT
To calculate the magnitude of the car's average acceleration, we need to find the change in velocity and divide it by the time taken.
Step 1: Convert the initial and final velocities to their vector components.
Initial Velocity (Vi) = 50.0 km/h at 60.0° north of east
Final Velocity (Vf) = 70.0 km/h at 40.0° south of east
Step 2: Break down the velocities into their x and y components.
Vi_x = Vi * cos(θ)
Vi_y = Vi * sin(θ)
Vf_x = Vf * cos(θ)
Vf_y = Vf * sin(θ)
Where θ is the angle measured from the positive x-axis.
For the initial velocity, we have:
Vi_x = 50.0 km/h * cos(60.0°) = 50.0 km/h * 0.5 = 25.0 km/h (eastward)
Vi_y = 50.0 km/h * sin(60.0°) = 50.0 km/h * 0.866 = 43.3 km/h (northward)
For the final velocity, we have:
Vf_x = 70.0 km/h * cos(40.0°) = 70.0 km/h * 0.766 = 53.6 km/h (eastward)
Vf_y = 70.0 km/h * sin(40.0°) = 70.0 km/h * 0.643 = 45.0 km/h (southward) [Note: Since south is the opposite direction of north, the y-component should be negative.]
Step 3: Calculate the change in velocity in the x and y directions.
ΔVx = Vf_x - Vi_x
ΔVy = Vf_y - Vi_y
ΔVx = 53.6 km/h - 25.0 km/h = 28.6 km/h (eastward)
ΔVy = 45.0 km/h - (-43.3 km/h) = 88.3 km/h (northward)
Step 4: Convert the time period to hours.
Time (t) = 30.0 minutes = 30.0 / 60.0 hours = 0.5 hours
Step 5: Calculate the average acceleration in the x and y directions.
Average Acceleration (Ax) = ΔVx / t
Average Acceleration (Ay) = ΔVy / t
Ax = (28.6 km/h) / (0.5 hours) = 57.2 km/h^2 (eastward)
Ay = (88.3 km/h) / (0.5 hours) = 176.6 km/h^2 (northward)
Step 6: Calculate the magnitude of the average acceleration.
Magnitude of Average Acceleration (A) = sqrt(Ax^2 + Ay^2)
A = sqrt((57.2 km/h^2)^2 + (176.6 km/h^2)^2)
A = sqrt(3283.84 + 31175.56)
A = sqrt(34459.40)
A ≈ 185.7 km/h^2
Therefore, the magnitude of the car's average acceleration is approximately 185.7 km/h^2.
To find the magnitude of the car's average acceleration, we first need to find the change in velocity (Δv) and the time taken (Δt).
The change in velocity is given by the final velocity (vf) minus the initial velocity (vi):
Δv = vf - vi
Let's calculate the change in velocity:
The initial velocity, vi = 50.0 km/h in the direction 60.0° north of east.
To break down the initial velocity into its x and y components, we use trigonometry. The x-component of the initial velocity, vix, can be found using the cosine function:
vix = vi * cos(theta)
where theta is the angle between the velocity vector and the x-axis.
vix = 50.0 km/h * cos(60.0°)
Similarly, the y-component of the initial velocity, viy, can be found using the sine function:
viy = vi * sin(theta)
viy = 50.0 km/h * sin(60.0°)
The final velocity, vf = 70.0 km/h in the direction 40.0° south of east.
To break down the final velocity into its x and y components, we use trigonometry. The x-component of the final velocity, vfx, can be found using the cosine function:
vfx = vf * cos(theta)
where theta is the angle between the velocity vector and the x-axis.
vfx = 70.0 km/h * cos(40.0°)
Similarly, the y-component of the final velocity, vfy, can be found using the sine function:
vfy = vf * sin(theta)
vfy = 70.0 km/h * sin(40.0°)
Now that we have the initial and final velocity components, we can calculate the change in velocity:
Δvx = vfx - vix
Δvy = vfy - viy
Next, we calculate the time taken, Δt. The time period is given as 30.0 minutes, but we need to convert it to hours for consistency with the velocity. There are 60 minutes in an hour, so:
Δt = 30.0 minutes / 60
Now we can calculate the average acceleration:
Average acceleration (a) = (Δvx / Δt)² + (Δvy / Δt)²
Finally, we find the magnitude of the average acceleration by taking the square root:
Magnitude of average acceleration = sqrt(a)
Let's plug in the values and calculate the magnitude of the car's average acceleration.