CALC. A car's velocity as a function of time is given by v_x(t) = α + βt^2, where α = 3.00 m/s and β = 0.100 m/s^3. (a) Calculate the average acceleration for the time interval t = 0 to t = 5.00 s.

Verified step by step guidance

1. Identify the given values: \( \alpha = 3.00 \, \text{m/s} \) and \( \beta = 0.100 \, \text{m/s}^3 \).
2. Write down the expression for the velocity as a function of time: \( v_x(t) = \alpha + \beta t^2 \).
3. Calculate the initial velocity at \( t = 0 \) by substituting \( t = 0 \) into the velocity equation: \( v_x(0) = \alpha + \beta \times 0^2 \).
4. Calculate the final velocity at \( t = 5.00 \, \text{s} \) by substituting \( t = 5.00 \, \text{s} \) into the velocity equation: \( v_x(5.00) = \alpha + \beta \times (5.00)^2 \).
5. Determine the average acceleration over the time interval from \( t = 0 \) to \( t = 5.00 \, \text{s} \) using the formula for average acceleration: \( a_{\text{avg}} = \frac{v_x(5.00) - v_x(0)}{5.00 - 0} \).

Key Concepts

Velocity as a Function of Time

In physics, velocity is defined as the rate of change of displacement with respect to time. When expressed as a function of time, such as v_x(t) = α + βt^2, it indicates how the velocity of an object changes over time. Here, α represents the initial velocity, while the term βt^2 shows how velocity increases with the square of time, reflecting non-linear acceleration.

Average Acceleration

Average acceleration is defined as the change in velocity over a specified time interval. It can be calculated using the formula a_avg = (v_final - v_initial) / (t_final - t_initial). In this case, to find the average acceleration from t = 0 to t = 5.00 s, one must evaluate the velocity function at these two time points and apply the formula to determine the average rate of change of velocity.

Calculus in Physics

Calculus is a branch of mathematics that deals with continuous change and is essential in physics for analyzing motion. In this context, the velocity function v_x(t) can be differentiated to find instantaneous acceleration, or integrated to find displacement. Understanding how to apply calculus concepts allows for deeper insights into the behavior of physical systems, such as the relationship between velocity, acceleration, and time.