A car is traveling at 100 km/h, due northwest. The driver puts on the brakes and turns the
corner. Four seconds later, he is heading east at 50 km/h. What is the average acceleration
To find the average acceleration, we need to calculate the change in velocity over the change in time.
First, let's determine the change in velocity. The initial velocity was 100 km/h due northwest, and the final velocity is 50 km/h due east. The change in velocity is the vector difference between the two velocities: 50 km/h due east - 100 km/h due northwest.
To add or subtract vectors, we need to resolve them into their horizontal and vertical components. In this case, the northwest direction can be broken down into equal horizontal and vertical components because it is at a 45-degree angle.
Using trigonometry, we can calculate the horizontal and vertical components of the initial velocity. The horizontal component is given by the formula: velocity * cos(theta), where theta is the angle between the velocity vector and the horizontal axis. Since the angle is 45 degrees, cos(45) is equal to 1/sqrt(2).
So, the horizontal component of the initial velocity is 100 km/h * 1/sqrt(2) = 100/sqrt(2) km/h. This component is parallel to the x-axis and remains unchanged when the car turns.
The vertical component of the initial velocity is also 100 km/h * 1/sqrt(2) = 100/sqrt(2) km/h. This component is in the opposite direction of the desired final velocity, so it is subtracted.
Therefore, the change in velocity is (50 km/h - 100/sqrt(2) km/h) due east in the x-axis and (100/sqrt(2) km/h) in the y-axis.
Next, let's determine the change in time. The problem mentioned that the driver took 4 seconds to make the turn.
Now, we can calculate the average acceleration. Average acceleration is defined as the change in velocity divided by the change in time.
To find the x-component of the acceleration, we divide the change in velocity in the x-direction by the change in time:
Ax = (50 km/h - 100/sqrt(2) km/h) / 4 s.
To find the y-component of the acceleration, we divide the change in velocity in the y-direction by the change in time:
Ay = (100/sqrt(2) km/h) / 4 s.
So, the average acceleration is (Ax, Ay) where Ax is the x-component and Ay is the y-component.