Here's how to determine the time at which the velocity of block C is zero, given the motion of collars A and B:
1. Given Information:
- Collar A starts from rest with a constant upward acceleration of \(\(3.6 \text{ in/s}^2\)\).
- Collar B moves downward with a constant velocity of \(\(20 \text{ in/s}\)\).
2. Understanding Velocities:
- At any time \(\( t \)\), the velocity of collar A, denoted as \(\( v_A \)\), can be calculated using the formula for uniformly accelerated motion:
\(\[
v_A = a_A \cdot t
\]\)
where \(\( a_A \)\) is the acceleration of collar A. Given \(\( a_A = 3.6 \text{ in/s}^2 \)\), we have:
\(\[
v_A = 3.6 \cdot t
\]\)
- The velocity of collar B, denoted as \(\( v_B \)\), is constant and downward. Given \(\( v_B = 20 \text{ in/s}\)\), we consider downward velocity as negative in the coordinate system where upward is positive. Thus:
\(\[
v_B = -20 \text{ in/s}
\]\)
3. Velocity of Block C:
- Block C's velocity is the algebraic difference between the velocities of collars A and B. Hence, the velocity of block C, denoted as \(\( v_C \)\), is given by:
\(\[
v_C = v_A - v_B
\]\)
- Substitute the known expressions:
\(\[
v_C = (3.6 \cdot t) - (-20)
\]\)
Simplifying, we get:
\(\[
v_C = 3.6 \cdot t + 20
\]\)
4. Finding the Time When \(\( v_C = 0 \)\):
- To determine when the velocity of block C is zero, we set \(\( v_C = 0 \)\):
\(\[
0 = 3.6 \cdot t + 20
\]\)
- Solving for \(\( t \)\):
\(\[
3.6 \cdot t + 20 = 0
\]\)
\(\[
3.6 \cdot t = -20
\]\)
\(\[
t = \frac{-20}{3.6}
\]\)
5. Final Calculation:
- Therefore, the time \(\( t \)\) at which the velocity of block C is zero is:
\(\[
t = -5.555555555555555 \text{ seconds}
\]\)
So, the time at which the velocity of block C is zero is \(\(-5.56\)\) seconds (rounded to two decimal places).
