Exploring Particle Motion with Integrals in AP Calculus AB

What does particle motion primarily connect in mathematical terms?

• Position, velocity, and acceleration

• Force, mass, and acceleration

• Velocity, time, and distance

• Speed, time, and displacement

What does a positive velocity indicate about the direction of an object?

• Moving to the right

• Moving upwards

• Stationary

• Moving to the left

How is instantaneous velocity determined?

• By calculating the total distance traveled

• By taking the derivative of the velocity function

• By using the absolute value of velocity

• By integrating the velocity function over time

What is the relationship between the signs of velocity and acceleration when speed is increasing?

• Signs are opposite

• No specific relationship

• Both signs are negative

• Both signs are positive

How do you calculate displacement using velocity?

• Take the average of the velocity over the time interval

• Differentiate the velocity function and evaluate at two points

• Integrate the velocity function from start to end time

• Sum the absolute values of velocity

What distinguishes total distance traveled from displacement?

• Total distance uses absolute values of velocity

• Displacement is not affected by direction

• Displacement considers only positive values

• Total distance is always zero

How is the actual position calculated in particle motion?

• Initial position plus the integral of velocity

• Average velocity times time plus initial position

• Total distance traveled plus initial position

• Initial position minus the integral of velocity

What does the integral of the absolute value of velocity represent?

• Average acceleration

• Displacement

• Total distance traveled

• Instantaneous speed

How can you estimate the derivative at a point using a table of values?

• By finding the maximum and minimum values

• By summing all the values and dividing by the number of points

• By integrating the values between two points

• By finding the slope between the closest points

What does a right Riemann sum help to approximate in the context of velocity?

• Instantaneous acceleration

• Average velocity

• Displacement

• Total distance traveled