To tackle the problem of a car's velocity as a function of time, we need to break it down into manageable parts. The velocity function is typically expressed as \( v(t) = \alpha + \beta t^2 \), where \( \alpha = 3.00 \, \text{m/s} \) and \( \beta = 0.100 \, \text{m/s}^3 \). Let's go through each part step by step.
Calculating Average Acceleration
The average acceleration over a time interval can be calculated using the formula:
Average Acceleration (a_avg) = (v_final - v_initial) / (t_final - t_initial)
First, we need to find the initial and final velocities:
- At \( t = 0 \):
v(0) = \( \alpha + \beta(0)^2 = 3.00 \, \text{m/s} \)
- At \( t = 5.00 \):
v(5) = \( \alpha + \beta(5)^2 = 3.00 + 0.100(25) = 3.00 + 2.50 = 5.50 \, \text{m/s} \)
Now, substituting these values into the average acceleration formula:
a_avg = (5.50 - 3.00) / (5.00 - 0) = 2.50 / 5.00 = 0.50 \, \text{m/s}^2
Finding Instantaneous Acceleration
Instantaneous acceleration is the derivative of the velocity function with respect to time:
a(t) = dv/dt
Given \( v(t) = \alpha + \beta t^2 \), we differentiate:
a(t) = d/dt(\alpha + \beta t^2) = 0 + 2\beta t = 2(0.100)t = 0.200t
Now, we can calculate the instantaneous acceleration at \( t = 0 \) and \( t = 5.00 \):
- At \( t = 0 \):
a(0) = 0.200(0) = 0 \, \text{m/s}^2
- At \( t = 5.00 \):
a(5) = 0.200(5) = 1.00 \, \text{m/s}^2
Visualizing the Motion: Velocity-Time Graph
To create a Velocity-Time graph for the car's motion from \( t = 0 \) to \( t = 5.00 \), we plot the calculated velocities at the specified times:
- At \( t = 0 \): \( v(0) = 3.00 \, \text{m/s} \)
- At \( t = 5.00 \): \( v(5) = 5.50 \, \text{m/s} \)
The graph will show a curve starting at \( (0, 3.00) \) and ending at \( (5.00, 5.50) \). Since the velocity function is quadratic, the graph will be a parabola opening upwards. You can sketch this by plotting the points and connecting them smoothly.
Summary of Results
- Average Acceleration: \( 0.50 \, \text{m/s}^2 \)
- Instantaneous Acceleration at \( t = 0 \): \( 0 \, \text{m/s}^2 \)
- Instantaneous Acceleration at \( t = 5.00 \): \( 1.00 \, \text{m/s}^2 \)
This analysis provides a comprehensive understanding of the car's motion over the specified time interval. If you have any further questions or need clarification on any part, feel free to ask!
