Question

The figure shows a cart of mass 10kg initially kept at rest on a frictionless surface. A jet of liquid of density 1000 kg/m³ is directed on the cart. Velocity of the jet is 10m/s and its cross section area is 2 × 10⁻⁴ m². If at t = 0, cart is released and allowed to move freely in horizontal direction then choose the CORRECT statement(s) :-

• A. Force on the cart at t = 0 s is 40 N
• B. Velocity of cart at t = 10 s is 8 m/s
• C. Acceleration of cart at t = 10 s is 1 m/s²
• D. Power supplied to the cart when cart is moving with velocity 5 m/s is 50 W.

Answer 1

Answer: (A)

A

Answer 2

Let the mass of the cart be $M = 10$ kg. Density of the liquid, $\rho = 1000$ kg/m³. Cross-sectional area of the jet, $A_0 = 2 \times 10^{-4}$ m². Velocity of the jet, $v_0 = 10$ m/s.

We assume the liquid jet is perfectly reflected by the cart. The force exerted by the jet on the cart is given by the rate of change of momentum of the liquid.

The rate of mass flow of the jet is: $\frac{dm}{dt} = \rho A_0 v_0 = (1000 \text{ kg/m}^3) \times (2 \times 10^{-4} \text{ m}^2) \times (10 \text{ m/s}) = 0.2$ kg/s.

For perfect reflection, the change in momentum of a mass $dm$ of the liquid is $dm(-v_0) - dm(v_0) = -2 dm v_0$. The force exerted by the jet on the cart is the rate of change of momentum of the liquid: $F = \frac{dm}{dt} (-v_0 - v_0) = -2 \frac{dm}{dt} v_0$. This is the force on the liquid. The force on the cart is the reaction force, $F_{cart} = -F = 2 \frac{dm}{dt} v_0$.

Alternatively, the force on the cart can be calculated as $F = 2 \rho A_0 v_0^2$. $F = 2 \times (1000 \text{ kg/m}^3) \times (2 \times 10^{-4} \text{ m}^2) \times (10 \text{ m/s})^2$ $F = 2 \times 1000 \times 2 \times 10^{-4} \times 100 = 40$ N.

Since the surface is frictionless, the acceleration of the cart is: $a = \frac{F}{M} = \frac{40 \text{ N}}{10 \text{ kg}} = 4 \text{ m/s}^2$.

This acceleration is constant because the force is constant.

Now let's evaluate the given statements:

(A) Force on the cart at t = 0 s is 40 N. With the assumption of perfect reflection, the force is constant at 40 N. This statement is CORRECT.

(B) Velocity of cart at t = 10 s is 8 m/s. Since the acceleration is constant ($a=4$ m/s²) and the initial velocity $v(0)=0$, the velocity at time $t$ is $v(t) = v(0) + at = 0 + 4t$. At $t=10$ s, $v(10) = 4 \times 10 = 40$ m/s. This statement is INCORRECT.

(C) Acceleration of cart at t = 10 s is 1 m/s². The acceleration is constant at 4 m/s². This statement is INCORRECT.

(D) Power supplied to the cart when cart is moving with velocity 5 m/s is 50 W. Power $P = F \times v$. With $F = 40$ N and $v = 5$ m/s: $P = 40 \text{ N} \times 5 \text{ m/s} = 200$ W. This statement is INCORRECT.

Therefore, only statement (A) is correct.