You drop a frog (initial velocity = 0) into a cart moving at 4 m/s. The frog has a mass of 1 kg. The cart has a mass of 3 kg. How fast are the cart and the frog going after the frog was dropped into the cart?

To determine the final velocity of the frog and the cart after the frog is dropped, we need to apply the principle of conservation of momentum.

Momentum is a vector quantity, and it is defined as the product of mass and velocity. The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, no external forces are mentioned, so the total momentum before and after the frog is dropped will be equal.

Before the frog is dropped, the cart is already moving with a velocity of 4 m/s, and the frog is at rest (velocity = 0). The masses of the frog and the cart are 1 kg and 3 kg, respectively.

Let's denote the final velocity of both the frog and the cart as Vc. According to the conservation of momentum, the initial momentum of the system (cart and frog together) will be equal to the final momentum:

Initial momentum = Final momentum
(3 kg) x (4 m/s) + (1 kg) x (0 m/s) = (3 kg + 1 kg) x Vc

Simplifying the equation:
12 kg m/s = 4 kg x Vc

Dividing both sides by 4 kg:
12 kg m/s ÷ 4 kg = Vc

Therefore, the final velocity of both the cart and the frog after the frog is dropped into the cart is 3 m/s.