Final answer:
The final velocity of the cart is approximately 0.857 m/s. The impulse exerted by the cart on the package is approximately -21.43 kg·m/s. The fraction of the initial energy lost in the impact is approximately 0.770.
Step-by-step explanation:
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
a) Using conservation of momentum, we know that the initial momentum of the package is equal to the final momentum of the cart and package system. The initial momentum of the package is given by:
p_initial = m_package * v_package = (10 kg) * (3 m/s) = 30 kg·m/s
The final momentum of the cart and package system is given by:
p_final = (m_cart + m_package) * v_final
Since the cart is initially at rest and can roll freely with no friction, the final velocity of the cart and package system will be the same. Therefore, we can set up the following equation:
p_initial = p_final
30 kg·m/s = (25 kg + 10 kg) * v_final
Simplifying the equation:
v_final = 30 kg·m/s / 35 kg = 0.857 m/s
Therefore, the final velocity of the cart is approximately 0.857 m/s.
b) The impulse exerted by the cart on the package is equal to the change in momentum of the package. We can calculate it using the equation:
Impulse = m_package * (v_final - v_package)
Impulse = 10 kg * (0.857 m/s - 3 m/s) = -21.43 kg·m/s
Therefore, the impulse exerted by the cart on the package is approximately -21.43 kg·m/s.
c) To determine the fraction of the initial energy lost in the impact, we can compare the initial kinetic energy of the package to the final kinetic energy of the cart and package system. The initial kinetic energy of the package is given by:
K_initial = (1/2) * m_package * (v_package)^2
K_initial = (1/2) * 10 kg * (3 m/s)^2 = 45 J
The final kinetic energy of the cart and package system is given by:
K_final = (1/2) * (m_cart + m_package) * v_final^2
K_final = (1/2) * (25 kg + 10 kg) * (0.857 m/s)^2 = 10.306 J
The fraction of the initial energy lost in the impact can be calculated as:
Fraction of energy lost = (K_initial - K_final) / K_initial
Fraction of energy lost = (45 J - 10.306 J) / 45 J = 0.770
Therefore, the fraction of the initial energy lost in the impact is approximately 0.770.