Attempting to stop on a slippery road, a car moving at 90 km/h skids across the road at a 30 angle to its initial motion, coming to a stop in 3.8s. Determine the average acceleration in m/s^2, using a coordinate system with the x-axis in the direction of the car's original motion and the y-axis toward the side of the road to which the car skids. Hence find ax and ay?
POOPOO
To find the average acceleration, we need to calculate the changes in velocity in both the x-axis and y-axis directions.
Given:
Initial velocity (u) = 90 km/h
Angle of skid (θ) = 30 degrees
Time (t) = 3.8 s
Step 1: Convert the initial velocity from km/h to m/s.
Using the conversion factor: 1 km/h = 1000/3600 m/s.
So, u = (90 * 1000) / 3600 = 25 m/s.
Step 2: Resolve the initial velocity into its x-component (vx) and y-component (vy).
vx = u * cos(θ) = 25 * cos(30) = 25 * (√3 / 2) = 25 * 0.866 ≈ 21.65 m/s.
vy = u * sin(θ) = 25 * sin(30) = 25 * (1/2) = 12.5 m/s.
Step 3: Determine the changes in velocity in both the x-axis and y-axis directions.
Using the equation: Δv = v - u, where v is the final velocity and u is the initial velocity.
Δvx = 0 - vx, because the car comes to a stop in the x-axis direction.
Δvx = -21.65 m/s.
Δvy = 0 - vy, because the car comes to a stop in the y-axis direction.
Δvy = -12.5 m/s.
Step 4: Calculate the average acceleration using the formula:
Average acceleration (a) = Δv / t.
ax = Δvx / t = (-21.65) / 3.8 ≈ -5.7 m/s².
ay = Δvy / t = (-12.5) / 3.8 ≈ -3.29 m/s².
Therefore, the average acceleration is approximately -5.7 m/s² in the x-axis direction and approximately -3.29 m/s² in the y-axis direction.