Questions: A turtle's velocity changes from v1=1.0 mm / s at Theta=0° to v2=1.2 mm / s at Theta=20°. What is the change in the turtle's velocity? Give the answer in component form (Δvx, Δvy).

A turtle's velocity changes from v1=1.0 mm / s at Theta=0° to v2=1.2 mm / s at Theta=20°. What is the change in the turtle's velocity? Give the answer in component form (Δvx, Δvy).

Transcript text: A turtle's velocity changes from $v_{1}=1.0 \mathrm{~mm} / \mathrm{s}$ at $\Theta=0^{\circ}$ to $v_{2}=1.2 \mathrm{~mm} / \mathrm{s}$ at $\Theta=20^{\circ}$. What is the change in the turtle's velocity? Give the answer in component form $\left(\Delta v_{x}, \Delta v_{y}\right)$.

Solution

Solution Steps

Step 1: Determine Initial Velocity Components

The initial velocity $ v_1 $ is given as $ 1.0 \, \text{mm/s} $ at $ \Theta = 0^\circ $.

The components of $ v_1 $ are: \[ v_{1x} = v_1 \cos(0^\circ) = 1.0 \, \text{mm/s} \] \[ v_{1y} = v_1 \sin(0^\circ) = 0 \, \text{mm/s} \]

Step 2: Determine Final Velocity Components

The final velocity $ v_2 $ is given as $ 1.2 \, \text{mm/s} $ at $ \Theta = 20^\circ $.

The components of $ v_2 $ are: \[ v_{2x} = v_2 \cos(20^\circ) = 1.2 \cos(20^\circ) \approx 1.2 \times 0.9397 = 1.1276 \, \text{mm/s} \] \[ v_{2y} = v_2 \sin(20^\circ) = 1.2 \sin(20^\circ) \approx 1.2 \times 0.3420 = 0.4104 \, \text{mm/s} \]

Step 3: Calculate Change in Velocity Components

The change in velocity components $ \Delta v_x $ and $ \Delta v_y $ are: \[ \Delta v_x = v_{2x} - v_{1x} = 1.1276 \, \text{mm/s} - 1.0 \, \text{mm/s} = 0.1276 \, \text{mm/s} \] \[ \Delta v_y = v_{2y} - v_{1y} = 0.4104 \, \text{mm/s} - 0 \, \text{mm/s} = 0.4104 \, \text{mm/s} \]

Step 4: Round to Four Significant Digits

Rounding the changes in velocity components to four significant digits: \[ \Delta v_x \approx 0.1276 \, \text{mm/s} \approx 0.13 \, \text{mm/s} \] \[ \Delta v_y \approx 0.4104 \, \text{mm/s} \approx 0.41 \, \text{mm/s} \]