Question

34 theme patterns the equations of motion can only be used when acceleration treated as constant. for comparison, sketch by hand or computer and label example velocity vs. time graphs for the two cases—constant accelerated motion and accelerated motion that is not constant.

Explanation:

1. Constant Accelerated Motion (e.g., Free Fall with Constant g):

• Velocity - Time Relationship: The velocity as a function of time is given by \( v(t)=v_0 + at \), where \( v_0 \) is the initial velocity and \( a \) (acceleration) is constant.
• Graph Shape: This is a straight line (linear graph) with:
• Slope = \( a \) (constant, so the line has a fixed steepness).
• Y - intercept = \( v_0 \) (velocity at \( t = 0 \)).
• Example: If an object is dropped (\( v_0=0 \)) and \( a = g=9.8\ m/s^2 \) (downward), the velocity graph is a straight line starting at (0, 0) and increasing with a slope of \( 9.8\ m/s^2 \).

1. Non - Constant Accelerated Motion (e.g., Car with Changing Acceleration):

• Velocity - Time Relationship: Acceleration \( a(t) \) is not constant, so velocity \( v(t)=\int a(t)dt + v_0 \) (the integral of a non - constant function).
• Graph Shape: This is a curved line (non - linear graph) because the slope (which represents acceleration \( a=\frac{dv}{dt} \)) changes over time.
• Example: A car that speeds up, then slows down, then speeds up again. The velocity - time graph will have a slope that increases, then decreases, then increases, resulting in a curve (e.g., a parabola if \( a(t) \) is quadratic, or a more complex curve for arbitrary \( a(t) \)).

Answer:

• Constant Accelerated Motion Velocity - Time Graph: A straight line with slope equal to the constant acceleration and y - intercept equal to the initial velocity.
• Non - Constant Accelerated Motion Velocity - Time Graph: A curved line where the slope (acceleration) changes with time.