Impulse: Definition, Formula, Impulse-Momentum Theorem

Impulse: Have you seen the breaking of wooden boards or bricks by a karate punch? What causes the bricks to break by a single punch? What is the reason behind this? IMPULSE! In breaking wooden boards or bricks by a karate punch, the collision between the hand and brick is only for a few milliseconds. The applied external force is large and the time of collision is short, the average impulsive force is relatively large. Thus when you break a wooden board or bricks by a karate punch, you have to apply significant force for the minimum time, which is impulse.

What is an Impulse?

The figure below shows a popular game often played by adults in the gaming zone, in which the contestant tries to see how high a weighted marker can be raised by hitting a target with a sledgehammer. What physical quantity does the device measure? Is it the average force, the maximum force, the work done, the impulse, the energy transferred, the momentum transferred, or something else?

Here, the device will measure impulse. The impulse of the net force acting on a particle during a given time interval is equal to the change in momentum of the particle during that interval. Since the contestant is hitting the target with a sledgehammer, the change in momentum is large, and the time of collision is short. Therefore, it signifies that the average impulsive force will be relatively large. Suppose two persons bring the harmer from the same height, but they are hitting with different forces. For the person who hits with greater force for the short time interval, the imparted impulse will be huge, and as a result, the height of the mark will be more. Thus the device will measure impulse.

Thus, Impulse is a quantity that describes the effect of a net force acting on an object (a kind of “moving force”). A large force acting for a short time to produce a finite change in momentum is called an impulsive force. In the history of science, impulsive forces were put in a conceptually different category from ordinary forces. Newtonian mechanics has no such distinction. Impulsive force is like any other force – except that it is large and acts for a short time

Formula to Calculate Impulse

An impulse of a force is defined as the change in momentum produced by force, and it is equal to the product of force and the time for which it acts. Thus, Impulse acts when a large force acts for a concise time duration to bring a finite change in the body’s momentum, and the force acting on the body is called impulsive force or force of impulse. Thus, the formula to calculate the impulse can be given as:

\({\rm{ Impulse }} = {\rm{ Force }} \times {\rm{ time }}\)

Unit of Impulse

Impulse is a term that quantifies the overall effect of a force acting over time. It is conventionally given the symbol \(\vec j\). Impulse is equal to the product of force and the time for which the force acts. Thus, the SI unit of impulse can be calculated by multiplying the units of force and time. The SI unit of force is newton \((\rm{N})\) and the SI unit of time is second \((\rm{s})\). Therefore,

\(\vec j = \vec F \cdot t\)

\( \Rightarrow \) The SI unit of impulse is Newton-second.

Dimensional formula of force is, \(\left[ {{M^1}{L^1}{T^{ – 2}}} \right]\)

Dimensional formula of time is, \(\left[ {{M^0}{L^0}{T^1}} \right]\)

Thus, the Dimensional formula of Impulse, \([\vec j] = \left[ {{M^1}{L^1}{T^{ – 1}}} \right]\)

Impulse is a vector quantity, and its direction is the same as the direction of \(\vec F\).

Impulse Equation

Newton’s second law of motion states that the rate of change of momentum of an object is directly proportional to the force applied. According to Newton’s second law of motion, if a body of mass \({\rm{m}}\) moving initially with velocity \(\vec u\) moving under acceleration \(\vec a\) achieves a final velocity \(\vec v\) in a small duration of time \({\rm{t}}\), then the impulsive force \(\vec F\) acting on it becomes:

\(\vec F = m\vec a = m\frac{{(\vec v – \vec u)}}{t}\)

\(\therefore \vec Ft = m\vec v – m\vec u = \vec j\)

where \(\vec j\) is the impulse generated due to the force.

The area under the Force-time curve gives the magnitude of impulse.

Impulse-Momentum Relation

The forces that we deal with in our day to day lives are variable quantity. Thus impulse will be,

\(\vec j = \int_{{t_1}}^{{t_2}} \vec Fdt\)

\(\vec F = \frac{{d\vec p}}{{dt}}\) (\(\vec p\) is the momentum)

\(\therefore \vec j = \int_{{t_1}}^{{t_2}} \frac{{d\vec p}}{{dt}}dt = {p_2} – {p_1} = \Delta p\)

\(\therefore \vec j = {p_2} – {p_1} = \Delta p\)

From the above formula, \({\rm{ Impulse }} = {\rm{ change\,in\,momentum }}\)

Momentum is a measurement of mass in motion: how much mass is in how much motion. It is usually given the symbol \(\vec p\)

By definition, momentum is equal to the product of mass and velocity of the given body; thus, \(\vec p = m\vec v\)

Where \({\rm{m}}\) is the mass and \(\vec v\) is the velocity. The SI unit for momentum is \({\rm{kg}} \cdot {\rm{m}}/{\rm{s}}\) and momentum is always a vector quantity.

Thus, the impulse is also a vector quantity, and its units will be:

\(\rm{kgms}^{ – 1} = \rm{kgms}^{ – 2} \rm{s} = \rm{Ns}\) (newton-second)

Hence, impulse has two different units; it can be measured in either kilogram meter per second \(({\rm{kgm}}/{\rm{s}})\) or Newton times seconds \({\rm{ (Ns) }}\).

Specific Impulse

Specific impulse is a measure of the efficiency of rocket fuels. It can be defined as impulse per mass (or thrust per mass flow rate), and it is equal to exhaust velocity and has the SI unit of meter per second.

\(j_{SP} = \frac{j}{m} = \frac{T}{{{q_m}}} = v\)

Specific impulse can also be defined as impulse per weight (or thrust per weight flow rate)
It is then equal to exhaust velocity divided by the acceleration due to gravity and has the SI unit of second.

\({j_{SP}} = \frac{j}{{mg}} = \frac{T}{{{q_m}g}} = \frac{v}{g}\)

A Common Phenomena

1. In baseball, when the player hits the ball, it receives a sudden change in momentum. This leads to a short reaction time and the development of a large impulsive force. 

2. Airbags in cars are designed to bear the impact of impulse. When a driver gets into an accident, their momentum carries them forward into the steering wheel. With an airbag installed in the car, a smaller force is exerted over an extended time duration, bringing a considerable change in its momentum, saving drivers from injury. Without the airbag, a large force is exerted in small duration, causing more damage to the driver.

3. A cricket player lowers his hand while catching the ball. This increases the time of impact and decreases the effect of force.

4. When someone falls from the bed on a cemented floor, they receive more injuries when compared to falling on a heap of sand. This happens because the sand yields more than the cemented floor, increasing the impact time and lowering the impact of force.

Solved Problems

Q.1. An object collides with a solid wall, and after the collision, it stops. Now. If the object’s weight was \(4.0\;{\rm{kg}}\) and the object travels with a velocity of \(20\;{\rm{m}}/{\rm{s}}\) before it hit the wall. Calculate the impulse of the object.
Solution: \(\Delta p = {p_f}\,-\,{p_i}\)
\(\Delta p = (4.0\;{\rm{kg}})(0\;{\rm{m}}/{\rm{s}}) – (4.0\;{\rm{kg}})(20\;{\rm{m}}/{\rm{s}})\)
\(\Delta p = – 80\;{\rm{kg}}\;{\rm{m}}/{\rm{s}}\)
Since Impulse \( = \) Change in momentum, thus Impulse \( = – 80\;{\rm{kg}}\;{\rm{m}}/{\rm{s}}\).

Summary

An impulse of a force is defined as the change in momentum produced by force, and it is equal to the product of force and the time for which it acts.

• 1. The formula to calculate the impulse can be given as: \({\rm{ Impulse }} = {\rm{ Force }} \times {\rm{ time }}\)
• 2. The SI unit of impulse is Newton-second.
• 3. Thus, the Dimensional formula of Impulse, \([\vec j] = \left[ {{M^1}{L^1}{T^{ – 1}}} \right]\)
• 4. Impulse is a vector quantity, and its direction is the same as the direction of \(\vec F\).
• If a body of mass \(m\) moving initially with velocity \(\vec u\) moving under acceleration \(\vec a\) achieves a final velocity \(\vec v\) in a small duration of time \(t\), then the impulsive force \(\vec F\) acting on it becomes: \(\vec F.t = m\vec v – m\vec u = \vec j\) , where \(\vec j\) is the impulse generated due to the force.
• 5. The area under the Force-time curve gives the magnitude of impulse.
• Impulse is equal to the change in momentum of the given object. \(\therefore \vec j = {p_2}\,-\,{p_1} = \Delta p\)
• Common examples of impulse are, hitting of a cricket ball by a batsman, catching of a ball etc.

Frequently Asked Questions

Q.1. What are human impulses?
Ans: According to biology, an electrical signal is travelling along the axon of a neuron. Nerve impulses excite or inhibit activity in other neurons or the body’s tissues, such as muscles. Impulse can be defined as a general desire or a sudden wish to eat chocolate or hear a song.

Q.2. What is an impulse? Explain with some real-life examples.
Ans: Impulse in Physics is a quantity that gives the effect of large force acting over small time to cause a greater change in the momentum of an object. It is equal to the product of force and the duration for which that force acts.
1. Golf: When the player hits the golf ball, it receives a sudden change in momentum. This leads to a short reaction time and the development of a large impulsive force.
2. Tennis: When a tennis ball hits the racket, it is supplied with a high magnitude and short duration force that helps change the ball’s direction of motion. The force exerted by the player acting on the ball is the impulsive force.

Q.3. What is linear momentum?
Ans: Momentum gives the measure of mass in motion and is defined as the product of the mass and velocity of an object, and it is a vector quantity.

Q.4. What is the difference between impulse and impulsive force?
Ans: An impulsive force is a force that acts on an object for a short period. For example, in a collision that results in a change in the velocity or momentum of one or more objects involved in the collision, an impulsive force is primarily produced. In contrast, an impulse is the effect of an impulsive force acting over a short duration; it gives the change in momentum of a body brought by a sudden force.

Q.5. What is the unit of impulse?
Ans: The SI unit of impulse is newton-second and also can be measured in kilogram-meter per second.

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