Human heart is pumping blood with constant velocity v m `"s"^(-1)` at the rate of M kg `"s"^(-1)`. The force required for this is (in N)

Options

A. M
B. Mv
C. `(M)/(v)`
D. `v("dM")/("dt")`

Answer

To find the force required for the human heart to pump blood at a constant velocity, we can use the principles of physics, particularly Newton's second law of motion. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- The velocity of blood being pumped is \( v \) m/s.
- The rate of blood flow (mass flow rate) is \( M \) kg/s.

### Step 2: Recall Newton's Second Law of Motion
Newton's second law states that the force \( F \) required to change the momentum of an object is equal to the rate of change of momentum. Mathematically, this is expressed as:
\[ F = \frac{dp}{dt} \]
where \( p \) is the momentum.

### Step 3: Define Momentum
The momentum \( p \) of an object is defined as the product of its mass \( m \) and its velocity \( v \):
\[ p = mv \]

### Step 4: Calculate the Rate of Change of Momentum
Since the mass flow rate is given as \( M \) kg/s, we can express the rate of change of momentum as:
\[ \frac{dp}{dt} = \frac{d(mv)}{dt} \]
Given that the velocity \( v \) is constant, we can simplify this to:
\[ \frac{dp}{dt} = v \frac{dm}{dt} \]
Here, \( \frac{dm}{dt} \) is the mass flow rate, which is equal to \( M \) kg/s.

### Step 5: Substitute the Values
Substituting \( \frac{dm}{dt} = M \) into the equation gives:
\[ \frac{dp}{dt} = v \cdot M \]

### Step 6: Write the Force Equation
Thus, the force \( F \) required to pump the blood is:
\[ F = v \cdot M \]

### Final Answer
The force required for the heart to pump blood is:
\[ F = Mv \text{ N} \]

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