A small cart of mass 300g when emty can move freely on a horizontal surface. The cart contains 100g of sand and moves initially with speed of 18cm s^-1.
a)what would be the speed of the cart if a further 200g of sand were dropped vertically into it from a stationary container?
a)If instead of adding sand to the cart,what would be the final speed of the cart?.
Another conservation of momentum question. mv before = mv after. Solve for second v.
To answer both questions, we need to apply the principle of conservation of momentum. Momentum is defined as the product of an object's mass and its velocity. According to the principle of conservation of momentum, the total momentum of a system remains constant if no external forces act upon it.
Let's break down the problem and calculate the final speed of the cart in each scenario:
a) When 200g of sand is dropped vertically into the cart:
1. Calculate the initial momentum (before the sand is dropped):
- Mass of the cart (empty): 300g = 0.3kg
- Velocity of the cart: 18cm/s = 0.18m/s
- Initial momentum = mass × velocity = 0.3kg × 0.18m/s = 0.054 kg⋅m/s
2. Calculate the momentum of the sand before it is dropped:
- Mass of the sand: 100g = 0.1kg
- Since the sand is in a stationary container, its initial velocity is 0.
- Momentum of the sand = mass × velocity = 0.1kg × 0 = 0 kg⋅m/s
3. Calculate the final momentum (after the sand is dropped):
- The mass of the cart now includes the added sand: 300g (cart) + 200g (sand) = 500g = 0.5kg
- The velocity of the cart is what we need to find.
- Final momentum = mass × velocity = 0.5kg × velocity
4. Apply the conservation of momentum: The sum of the initial momenta is equal to the sum of the final momenta:
- Initial momentum (cart) + Initial momentum (sand) = Final momentum (cart)
- 0.054 kg⋅m/s + 0 kg⋅m/s = 0.5kg × velocity
Now, solve the equation for the final velocity (v) of the cart:
0.054 kg⋅m/s + 0 kg⋅m/s = 0.5kg × velocity
0.054 kg⋅m/s = 0.5kg × velocity
velocity = 0.054 kg⋅m/s / 0.5kg
velocity = 0.108 m/s
Therefore, the speed of the cart when 200g of sand is dropped vertically into it would be 0.108 m/s.
b) If instead of adding sand to the cart:
Since no external forces are acting on the cart, the total momentum of the system will remain constant.
We already calculated the initial momentum of the cart in part (a) as 0.054 kg⋅m/s.
For the final speed of the cart, there are no additional momentum contributions. We can assume the mass of the cart remains the same, and hence the final speed of the cart will also be 0.054 kg⋅m/s.
Therefore, if instead of adding sand to the cart, the final speed of the cart would be 0.054 m/s.