Impulse and Momentum: Calculating Impulse from Force vs. Time Graphs

Impulse and Momentum

Impulse with Variable Forces

Impulse is a fundamental concept in physics that describes the effect of a force acting over a period of time. When the force applied to an object varies with time, the impulse can be calculated using the area under a force vs. time graph. This section explains how to determine impulse from such graphs and relates it to changes in momentum.

• Impulse (J): The product of force and the time interval over which it acts. For variable forces, impulse is the area under the force vs. time graph.

• Momentum (p): Defined as the product of mass and velocity: $p = m v$

• Relationship: Impulse equals the change in momentum: $J = F \Delta t = \Delta p = m v_f - m v_0$

• Area Calculation: For simple shapes under the graph:

  • Rectangle: $A = b \ast h$

  • Triangle: $A = \frac{1}{2} b \ast h$

• Sign of Impulse: Areas above the time axis represent positive impulse; areas below represent negative impulse.

Example: Calculating Impulse from a Force vs. Time Graph

Consider a remote-controlled toy car moving along the x-axis, with a changing force applied as shown in a force vs. time graph. To find the impulse delivered to the car:

• Calculate the area under the force vs. time curve.

• If the car has a mass of 2 kg and starts from rest, use the impulse to find the final speed:

Formula:

$J = m v_f - m v_0$

Since $v_0 = 0$ (starts from rest):

$v_f = \frac{J}{m}$

Work vs. Impulse: Area Under Force Graphs

The area under a force vs. displacement (F vs. x) graph gives the work done, while the area under a force vs. time (F vs. t) graph gives the impulse.

• Work: $W = F \Delta x$

• Impulse: $J = F \Delta t$

Problem: Finding Maximum Force for a Given Impulse

Given a force vs. time graph, determine the value of maximum force ($F_{max}$) required to achieve a specified impulse (e.g., 6 N·s). This involves calculating the area under the curve and setting it equal to the desired impulse.

Example: Baseball Bat and Ball

When a baseball bat hits a ball, a large force is exerted over a very short time. The impulse delivered can be calculated from the force vs. time graph. If the ball is initially at rest and has a mass of 200 g, the final speed after the impulse is:

• Calculate impulse from the area under the graph.

• Use $v_f = \frac{J}{m}$ to find the final speed.

Summary Table: Area Under Force Graphs

Graph Type	Area Represents	Significance
Force vs. Displacement (F vs. x)	Work	Energy transferred to/from object
Force vs. Time (F vs. t)	Impulse	Change in momentum

Key Equations

• $p = m v$

• $J = F \Delta t = \Delta p = m v_f - m v_0$

• $A_{rectangle} = b \ast h$

• $A_{triangle} = \frac{1}{2} b \ast h$

• $W = F \Delta x$