Question

A motor scooter rounds a curve on the highway at a constant speed of $20.0 \mathrm{m} / \mathrm{s} .$ The original direction of the scooter was due east; after rounding the curve the scooter is heading \(36^{\circ}\) north of east. The radius of curvature of the road at the location of the curve is $150 \mathrm{m}$ What is the average acceleration of the scooter as it rounds the curve?

Short answer

Answer: To find the average acceleration of the motor scooter, follow these steps: 1. Find the change in velocity vector by subtracting the initial velocity vector from the final velocity vector. 2. Break down the final velocity vector into its x and y components. 3. Calculate the magnitude of the change in velocity vector. 4. Determine the time taken to round the curve using the scooter's speed and the given radius of curvature. 5. Calculate the average acceleration using the formula $$a_{avg}=\frac{|\Delta v|}{\Delta t}$$.

Step-by-step solution

Find the change in velocity

The scooter's initial velocity is \(20.0\,m/s\) towards the east. After turning, it is heading \(36^\circ\) north of east with the same speed. We'll subtract the initial velocity vector from the final velocity vector to find the change in velocity vector. Let's represent the initial velocity vector as \(\vec{v_i}\) and the final velocity vector as \(\vec{v_f}\). Then, we can find the change in velocity vector, \(\Delta \vec{v}\), using: $$\Delta \vec{v}=\vec{v_f}-\vec{v_i}$$

Break down the final velocity vector

The final velocity, \(20.0\,m/s\) north of east, can be represented using its components in the x and y directions: \(x\)-component: $$v_{fx}=20\,\text{m/s} \cos 36^\circ$$ \(y\)-component: $$v_{fy}=20\,\text{m/s} \sin 36^\circ$$

Calculate the change in velocity vector

Now we can subtract the initial velocity vector from the final velocity vector component-wise: $$\Delta v_x=v_{fx}-v_i$$ $$\Delta v_y=v_{fy}$$ The magnitude of the change in velocity can be calculated as follows: $$|\Delta v|=\sqrt{(\Delta v_x)^2+(\Delta v_y)^2}$$

Find the time taken to round the curve

To find the time taken to cover the curved path, we'll use the formula: $$t=\frac{s}{v}$$ Where \(t\) is the time taken, \(s\) is the distance (length of the curve), and \(v\) is the speed. The length of the curve can be calculated using the given radius of curvature, \(R\): $$s=\text{arc length}=\text{road angle}\times R$$ Here, the road angle can be found using the formula: $$\text{road angle}=\frac{180^\circ\times s}{R\pi}$$ Once we have the time taken, we can move on to the final step.

Calculate the average acceleration

Now that we have the magnitude of the change in velocity and the time taken to round the curve, we can find the average acceleration using the formula: $$a_{avg}=\frac{|\Delta v|}{\Delta t}$$ This will provide us with the average acceleration of the motor scooter as it rounds the curve.