Question

a bicyclist, initially at rest, speeds up steadily. during this time interval she averages 4m/s. determine all unknowns. s_i = 0 m/s, s = 4 m/s, s_f = 2 m/s

Explanation:

Step1: Recall average speed formula

For uniformly accelerated motion (speeding up steadily, so constant acceleration), the average speed \( s \) is given by the formula \( s=\frac{s_i + s_f}{2} \), where \( s_i \) is the initial speed and \( s_f \) is the final speed.
We know \( s_i = 0 \, \text{m/s} \) (initially at rest) and \( s = 4 \, \text{m/s} \).

Step2: Solve for final speed \( s_f \)

Substitute the known values into the formula:
\( 4=\frac{0 + s_f}{2} \)
Multiply both sides by 2: \( 4\times2=0 + s_f \)
So, \( s_f = 8 \, \text{m/s} \). Wait, but in the given image, \( s_f \) is marked as 2? Wait, maybe I misread. Wait, no, let's check again. Wait, the problem says "speeds up steadily", initial speed \( s_i = 0 \), average speed \( s = 4 \). The formula for average speed in uniform acceleration is \( s=\frac{v_i + v_f}{2} \). So \( 4=\frac{0 + v_f}{2} \), so \( v_f = 8 \). But the yellow box has 2, maybe that's a mistake? Wait, no, maybe the variables are different. Wait, maybe \( s_i \) is initial, \( s \) is average, \( s_f \) is final. Wait, let's re-express.

Wait, the user's image shows \( s_i = 0 \), \( s = 4 \), and \( s_f \) is to be found? Wait, no, the yellow box has 2, but according to the formula, it should be 8. Wait, maybe the problem is different. Wait, maybe it's a typo, but let's proceed with the formula.

Wait, the correct formula for average velocity (speed, since direction is constant) in uniformly accelerated motion (constant acceleration) is \( \bar{v}=\frac{v_i + v_f}{2} \). Given \( v_i = 0 \), \( \bar{v}=4 \), so:

\( 4=\frac{0 + v_f}{2} \)

Multiply both sides by 2: \( v_f = 8 \, \text{m/s} \).

But in the image, the yellow box is 2, which is incorrect. So the correct final speed \( s_f \) should be 8 m/s.

Answer:

The final speed \( s_f \) is \( \boldsymbol{8} \, \text{m/s} \) (correcting the error in the given image's yellow box value, as per the average speed formula for uniformly accelerated motion).