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Calculus-Based Physics Introduction

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Calculus-Based Physics Introduction

A lab team is testing a cart that moves along a straight track. Use derivatives to describe how its motion changes and integrals to connect motion over time. Show units and enough work to make your reasoning clear.

Motion as a function

For the first test, the cart’s position is x(t) = 2t³ − 9t² + 12t metres, where t is measured in seconds and 0 ≤ t ≤ 4.

1.Differentiate x(t) to find the velocity function v(t). Include units.

2.Differentiate v(t) to find the acceleration function a(t). What is the cart’s acceleration at t = 2 s?

3.At what times is the cart momentarily at rest? Use v(t) = 0, and state whether each time is within the test interval.

4.Use the number line to identify where v(t) is positive or negative. In each interval, is the cart moving in the positive or negative x-direction?

Read the velocity graph

A second cart has the velocity-time graph shown. Between each pair of plotted points, velocity changes linearly. Positive signed area under the graph represents displacement in the positive direction.

Velocity-time graph: points (0 s, 0 m/s), (2 s, 4 m/s), and (5 s, −2 m/s). Horizontal axis: time t (s), from 0 to 5. Vertical axis: velocity v (m/s), from −2 to 4.

5.Find the acceleration from t = 0 to 2 s and from t = 2 to 5 s. Give each value in m/s².

6.At what time does the second cart reverse direction? Explain how the graph shows this.

7.Calculate the second cart’s net displacement from 0 to 5 s by finding the signed area under the velocity-time graph. Then calculate its total distance travelled.

Connect the calculus

Think of differentiation as finding an instantaneous rate of change and integration as accumulating change over time.

8.Which statement correctly connects position, velocity, and acceleration?

Velocity is the derivative of position, and acceleration is the derivative of velocity.
Position is the derivative of velocity, and velocity is the derivative of acceleration.
Velocity is the integral of position, and acceleration is the integral of velocity.
Position, velocity, and acceleration are independent quantities.

9.The second cart starts at x(0) = 3 m. Use its net displacement from Question 7 to find x(5). Then explain why this answer is not the same as the total distance travelled.

10.In one or two sentences, explain what the derivative of velocity means physically and what the signed area under a velocity-time graph represents.