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Impulse formula in classical mechanics is the integral of force F, over the time interval t, due to which it acts. Simply, impulse is the product of force and the time interval during which the force is applied. It is a measure of the change in momentum of an object that occurs when a force is applied to it for a certain amount of time. Mathematically, impulse can be represented by:
| Impulse = Force × (final time – initial time) → Impulse = Force × Δt → I = F × Δt |
Momentum is a physical quantity that quantifies the motion contained by a moving object, whereas impulse is a physical quantity that is utilized to gain an understanding of the change in the dynamical state of a moving particle.
The change in momentum induced by a very brief interaction of force with an object is also known as an impulse. We can use impulse once in a while or on a regular basis. For example, it is a kind of principle that we use when we hit a ball while playing.
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Key Terms: Momentum, Impulse, Velocity, Mass, Speed, Vector, Linear Impulse, Impulse-momentum Theorem, Conservation of Momentum
What is Momentum?
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Momentum is a measure of the motion of an object and is the product of its mass and velocity. It represents how difficult it is to stop or change the direction of the object's motion, and is a conserved quantity, meaning that in a closed system, the total momentum remains constant.
- The momentum of an object is considered to be a vector quantity, meaning that it has both a magnitude and a direction.
- Momentum is conserved in a closed system. This is because the total momentum of a system remains constant unless acted upon by an external force.
- This principle is known as the law of conservation of momentum.
- Momentum simply defines the measure of strength.
- An object which is considered stationary has zero momentum.
- Note that a slow-moving yet large object comes with large momentum. Similarly, a small yet fast-moving object also comes with large momentum.
- For example, If a Bowling ball and a Ping-Pong ball have the same velocity, the Bowling ball will have more momentum than the Ping-Pong ball since it is larger.
Momentum Formula
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The factors of mass and velocity influence momentum. In terms of an equation, an object's momentum is equal to its mass multiplied by its velocity. Mass multiplied by velocity equals momentum. As a result, the equation above may be rewritten as:
| \(\vec{p} = m\ \times \ \vec{v}\) |
Where,
- \(\vec{p}\) represents momentum
- m represents mass
- \(\vec{v}\) represents the velocity
The equation illustrates that an object's momentum is proportional to its mass and velocity. Additionally, momentum is a vector equal to the product of mass and velocity (also a vector). The SI unit of momentum is the kg.m/s.
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What is Impulse?
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The term "impulse" refers to a physical quantity that is used to estimate the change in a moving particle's dynamical state.
- It is the change in momentum of a body when it is subjected to a force for a short period of time.
- It is denoted by the letter ‘J’ and is a vector quantity.
- The inverse of a force is an impulse.
- It signifies that the magnitude of the impulse formula is determined by the force and time change.
- Impulse is measured in newton seconds (Ns) or kg m/s.
- The concept of impulse is useful for understanding the effect of a force acting on an object over a period of time.
- For example, in a car crash, the impulse of the collision determines the severity of the impact on the occupants of the car.
Also Read: Uniform Motion
Impulse Formula
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Impulse is the product of the force applied on an object and the time for which the force is known to act. The formula of Impulse is:
Impulse = Force × (final time – initial time)
→ Impulse = Force × Δt
Impulse Derivation
This formula is important because it relates the change in momentum of an object to the force applied to it. Thus,
| J = F × Δt |
This is the Linear Impulse Formula. Here,
- J = impulse
- F is the force of the object
- Δt is the change in time
As the impulse is a measure of how much momentum changes over time as a result of a force acting on it. Furthermore, there is an alternative formula for impulse.
| Impulse = Δp = pfinal – pinitial |
where
- Δp = change in momentum
- p initial = initial momentum
- p final = final momentum
Most importantly, the formula connects impulse to the object's change in momentum. In addition, impulse can be measured in kilograms meters per second (kg m/s) or Newton times seconds (Ns).
Dimensional Formula of Impulse
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We know that the physics equation for impulse is:
→ J = F × Δt …. (1)Here,
[M1L1T-2] is the dimensional formula for F, which is measured in Newton.-----(2)
[M0L0T1] is the dimensional formula for Δt, which is measured in seconds.-------(3)
By substituting the values of equations (2) and (3) into equation (1), we obtain the dimensional formula for impulse:
= [M1L1T-2] x [M0L0T1]
The following dimensional formula is obtained by solving the above equation as [M1L1T-1].
So, the needed dimension of the impulse formula is [M1L1T-1].
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Impulse-Momentum Equation
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The impulse-momentum formula is derived from the impulse-momentum theorem, which asserts that an object's change in momentum equals the impulse supplied to it. Impulse can be calculated by using the following impulse-momentum equation:
| J = m x Δv |
Simply, Impulse = m (Mass) x [v2(Velocity2) – v1(Velocity1)]
Where velocity1 and velocity2 are the velocities at two different points in a given time span. In other terms, it can be said as the change in velocity.
According to the preceding equation, the unit of impulse can be determined as follows:
As, Mass = in kg, Velocity = in m/s
So, J = kg . m/s or N.s
Thus,
| Conditions Required | |
|---|---|
| If Mass = Constant | FΔt = mΔv |
| If Mass Varies | Fdt = mdv + vdm |
Things to Remember
- The term "momentum" refers to the measurement of an object's motion.
- The amount of force necessary to create a specified change in momentum in a body over a given period of time is defined as an impulse.
- Impulse is a vector quantity measured in kilograms meters per second (kg m/s) or Newton times seconds (Ns) and has dimensional formula as [M1L1T-1].
- The measure of how much a force changes the momentum of an item when it operates on it for a brief period of time is called impulse. In this way, impulse and momentum are linked.
- The impulse-momentum formula is based on the impulse-momentum theorem, which specifies the object's change in momentum equals the applied impulse.
- In the following equations, J = F x Δt and J = m x Δv, the unit and the dimensional formula for impulse remain the same.
Also Read: Difference Between Scalar and Vector Quantities
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Sample Questions
Ques: When an object collides with a wall, it bounces back. Furthermore, the object's mass is 2.0 kg and its velocity is 10 m/s before it hits the wall. Furthermore, after it hits the wall, its velocity drops to -10 m/s (it is negative because it has bounced back in the opposite direction). Now compute the object's impulse. (2 marks)
Ans: As Δp = p final - p initial
Δp = m* v final - m * v initial
Δp = 2.0 * (-10) - 2.0 * 10
Δp = -20 kg m/s - (+ 20 kg m/s)
Δp = -40 kg m/s (Ans.)
Ques: A golfer smashes a 45-gram ball at a pace of 40 meters per second. For three seconds, the golf club is in touch with the ball. Calculate the average force the club exerts on the ball. (3 marks)
Ans: Mass (m) = 0.045 kg
Initial Velocity (vi) = 0
Final Velocity (vf) = 40m/s
Time = 0.003 s
Impulsive force (F) = m * a = m * [(vf – vi)/t]
= 0.045 * [(40-0)/0.003]
= 600N (Ans.)
Ques: A 5-gram ball falls to the ground after a free fall from a height. Gravitational acceleration, g = 10 ms-2. The ball's velocity before the collision is 6 ms-1, and the ball is reflected upright at 4 m/s after the collision. Calculate the impulse. (3 marks)
Ans: Mass of the ball (m) = 5 gm = 0.005 kg
Before the collision, the velocity of ball (vo) = -6 m/s
After the collision, the velocity of ball (vt) = 4 m/s
The direction before the collision is opposite the direction after the contact, as indicated by the plus and minus signs.
Since we need to find out Impulse (J), so the solution will be
Impulse (J) = Total change in momentum (Δp)
J = Δp = m * vt – m * vo
= m * (vt – vo)
J = (0.005)(4 – (-6)) = (0.005)(4 + 6) = (0.005)(10) = 0.05 N.s (Ans.)
Ques: As seen in the graph, a constant net force acts on a 2-kg mass with a starting velocity of 5 m/s. During the 5-second time, what is the impulse acting on the mass? What is the mass's final velocity after the 5-s interval? (2 marks)
Ans: After 5 s, The impulse would be equal to the rectangle's area.
So total impulse will be equal to total area = (10 N) * (5 s) = 50 N.s
As we know that Impulse = change in momentum
So, m * Δv = m * (vf – vi)
50 N.s = (2 kg) * (vf − 5 m/s)
vf = 30 m/s (Ans.)
Ques: When an object collides with a solid wall, it comes to a halt. Now. If the object's weight is 2.0 kg and it moves at a speed of 10 m/s before colliding with the wall. Calculate the object's impulse. (2 marks)
Ans: As Δp = p final - p initial
Δp = m* v final - m * v initial
Δp = 2.0 * (0) - 2.0 * 10
Δp = 0 kg m/s - 20 kg m/s
Δp = -20 kg m/s
Ques: For a 5-kg mass traveling horizontally, a graph of net force vs time is given. What is the mass's final velocity after 20 seconds if it starts at rest? (3 marks)
Ans: After 20 seconds, we must determine the mass's final velocity.
To get that velocity, however, we must first determine the mass's impulse. Because impulse equals force multiplied by time, this impulse equals the area under the graph in this situation (which is in the shape of a triangle during the 20-s interval).
So here, Impulse will be = (1/2) * (20) * (50) = 500 N.s
Now the final velocity can be now calculated as,
Impulse = change in momentum
So, 500 N.s = (5 kg) * (vf − 0 m/s)
vf = 100m/s (Ans.)
Ques: A batter strikes a ball straight at the bowler while keeping its initial 12 m/s speed. Calculate the impulse transferred to the ball if its mass is 0.15kg. (2 marks)
Ans: Given vi (Initial Velocity) = 12 m/s
vf (Final Velocity) = -12m/s
m (mass) = 0.15kg
Impulse is articulated as J = m * vf – m * vi
= m (vf – vi)
= 0.15 Kg (-12 -12) m/s = -3.6N.s
So, impulse = -3.6N.s (Ans.)
Ques: A 20-gram object was tossed to the left at a velocity of 4 m.s-1. The ball is reflected the right at a velocity of v2 = 2 m.s-1 after impacting with the wall. Calculate the impulse. (4 marks)
Ans: Mass of the ball (m) = 20 gram = 0.020 kg
Before collision, velocity of ball (vo) = -4 m/s (to the left)
After collision, velocity of ball (vt) = +2 m/s (to the right)
The direction before the collision is opposite the direction after the contact, as indicated by the plus and minus signs.
Since we need to find out Impulse (J), so the solution will be
Impulse (I) = the change in momentum (Δp)
= m * vt – m * vo
= m * (vt – vo)
= 0.02 (2 – (-4))
= 0.02 (2 + 4) = 0.02 (6)
= 0.12 N.s (Ans.)
Ques: How is momentum and impulse related to one another? (5 marks)
Ans: Momentum and impulse are closely related to each other in physics. When a force is applied to an object, it produces an impulse on the object. This impulse causes a change in the object's momentum.
In mathematical terms, the impulse is defined as the product of the force applied to an object and the time duration for which the force acts. Mathematically, it can be expressed as:
Impulse = Force x (Final Time – Initial Time)
However, momentum is defined as the product of the mass of an object and its velocity. Mathematically, it can be expressed as:
Momentum = Mass x Velocity
When an impulse is applied to an object, it causes a change in the object's momentum. This change in momentum can be calculated using the impulse-momentum theorem, which states that the impulse acting on an object is equal to the change in its momentum. Mathematically, this can be expressed as:
Impulse = Change in Momentum
or
FΔt = mΔv
where F is the force applied, Δt is the time duration for which the force acts, m is the mass of the object, and Δv is the change in the object's velocity.
Therefore, by applying a force for a certain time, we can change an object's momentum. This proves how both are related to each other.
Ques: What is the formula and unit of impulse? (2 marks)
Ans: Impulse is the product of force and time, and its formula is:
Impulse = Force x (Final Time – Initial Time)
The SI unit of impulse is the Newton-second (Ns) or kilogram-meter per second (kg·m/s). It can also be expressed in terms of the pound-second (lb·s) in the English system.
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