Here,
J=M¹L¹/T¹
since delta is dimensionless so mv=M¹L¹/T¹
hence, J=m × delta × v

Impulse and Change in Velocity

To understand the relationship between impulse and change in velocity, let's first define what impulse and change in velocity mean.

Impulse:
Impulse is a measure of the change in momentum of an object. It is defined as the product of the force acting on the object and the time for which the force acts. Mathematically, impulse (J) is given by the equation:

J = F * Δt

where F is the force acting on the object and Δt is the time interval for which the force acts.

Change in Velocity:
Change in velocity refers to the difference between the final velocity (vf) and the initial velocity (vi) of an object. Mathematically, change in velocity (Δv) is given by the equation:

Δv = vf - vi

where vf is the final velocity and vi is the initial velocity.

Deriving the Relationship:

To show that J = mΔv, we can use Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of momentum of the object. Mathematically, this can be expressed as:

F = Δp/Δt

where F is the force, Δp is the change in momentum, and Δt is the time interval.

Since momentum (p) is defined as the product of mass (m) and velocity (v), we can rewrite the equation as:

F = mΔv/Δt

Now, we can rearrange the equation to solve for impulse (J):

J = F * Δt
= mΔv/Δt * Δt
= mΔv

Therefore, we have shown that J = mΔv, where J is the impulse acting on a body of mass m and Δv is the change in velocity.

Conclusion:

In conclusion, the relationship between impulse and change in velocity is given by the equation J = mΔv. This equation is derived from Newton's second law of motion, which relates force, momentum, and time. Impulse is a measure of the change in momentum, and the change in velocity refers to the difference between the final and initial velocities of an object.