A cart loaded with 50kg of sand moving along a smooth surface with velocity 20m/s at a point of time sand starts falling at rare of 2kg/s Find the velocity of the cart after 100 second

To determine the velocity of the cart after 100 seconds, we can apply the principle of conservation of momentum. This principle states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In this case, the cart is losing mass as sand falls off, but no external horizontal forces are acting on it.

Initial Conditions

Let's break down the problem step by step:

• Initial mass of the cart (m₀): 50 kg
• Initial velocity of the cart (v₀): 20 m/s
• Rate of sand falling (r): 2 kg/s
• Time duration (t): 100 seconds

Calculating the Mass Loss

First, we need to find out how much sand has fallen off the cart after 100 seconds:

Mass of sand lost = Rate of sand falling × Time = 2 kg/s × 100 s = 200 kg

Since the cart initially had 50 kg of sand, it will run out of sand before 100 seconds are up. In fact, the cart will be empty after 25 seconds (50 kg / 2 kg/s).

Momentum Conservation Before Sand Runs Out

For the first 25 seconds, the cart maintains its momentum. The initial momentum (p₀) can be calculated as:

p₀ = m₀ × v₀ = 50 kg × 20 m/s = 1000 kg·m/s

Final Conditions After 25 Seconds

After 25 seconds, the cart has lost all its sand:

• Mass of the cart (m_f): 0 kg (since it started with 50 kg and lost all the sand)
• Velocity of the cart (v_f): We need to find this.

Applying Momentum Conservation

At the moment the cart runs out of sand, the momentum must still be conserved. However, since the mass of the cart becomes zero, we can only consider the momentum up to that point. After 25 seconds, the cart will continue moving at the last velocity it had before losing all the sand. Therefore, we need to calculate the velocity just before it runs out of mass.

Velocity Calculation

Using the conservation of momentum for the first 25 seconds:

p_final = p_initial = 1000 kg·m/s

Since the mass is still 50 kg at this point, we can find the velocity:

v_final = p_initial / m₀ = 1000 kg·m/s / 50 kg = 20 m/s

After 25 Seconds

After 25 seconds, the cart continues to move at 20 m/s, but now it has no mass to lose. Therefore, for the remaining 75 seconds, the velocity remains constant at 20 m/s because there is no further mass loss and no external forces acting on it.

Final Result

Thus, the velocity of the cart after 100 seconds remains:

Velocity of the cart after 100 seconds: 20 m/s