Particle Motion: Finding Velocity from Acceleration
The problem asks for the velocity of a particle at the specific instant when its acceleration becomes zero. We are given the initial condition that the particle starts from rest.
1. Define Acceleration and Initial Conditions
- Acceleration $a(t)$ is given by: $a(t) = (6 - t/4) \text{ cm/s}^2$.
- The particle starts from rest, meaning its initial velocity $v(0) = 0$.
2. Find Time When Acceleration is Zero
Set the acceleration function to zero and solve for time ($t$):
$a(t) = 0$ $6 - \frac{t}{4} = 0$ $6 = \frac{t}{4}$ $t = 6 \times 4 = 24$ seconds.The acceleration becomes zero after 24 seconds.
3. Determine Velocity Function using Integration
Velocity $v(t)$ is the integral of acceleration $a(t)$ with respect to time:
$v(t) = \int a(t) dt = \int (6 - \frac{t}{4}) dt$ $v(t) = 6t - \frac{t^2}{4 \times 2} + C$ $v(t) = 6t - \frac{t^2}{8} + C$Use the initial condition $v(0) = 0$ to find the constant $C$:
$0 = 6(0) - \frac{0^2}{8} + C$ $C = 0$Therefore, the velocity function is:
$v(t) = 6t - \frac{t^2}{8}$4. Calculate Velocity at $t = 24$s
Substitute $t = 24$ seconds into the velocity function:
$v(24) = 6(24) - \frac{(24)^2}{8}$ $v(24) = 144 - \frac{576}{8}$ $v(24) = 144 - 72$ $v(24) = 72 \text{ cm/s}$5. Round Off the Result
Rounding the velocity to one decimal place gives $72.0$ cm/s.
The calculated velocity $72.0$ cm/s falls within the range specified by the correct answer (71 to 73).
Important Questions from Indefinite Integrals
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