A 4.5-kg ham is thrown into a stationary 15-kg shopping cart. At what speed will the cart travel if the ham had an initial speed of 2.2 m/s? If the ham hits the cart with 2N of force, how long did the collision between the two last?
To find the final velocity of the shopping cart, we can use the conservation of momentum principle.
First, let's find the initial momentum of the ham and the shopping cart separately.
Initial momentum of ham (p_ham) = mass_ham * initial_velocity_ham
p_ham = 4.5 kg * 2.2 m/s
p_ham = 9.9 kg*m/s
Initial momentum of shopping cart (p_cart) = mass_cart * initial_velocity_cart
p_cart = 15 kg * 0 m/s (since the cart is initially stationary)
p_cart = 0 kg*m/s
According to the conservation of momentum principle, the total momentum before the collision (p_total_before) is equal to the total momentum after the collision (p_total_after).
p_total_before = p_ham + p_cart
Since the cart is initially stationary, its initial momentum is zero.
p_total_before = p_ham
p_total_before = 9.9 kg*m/s
The final momentum after the collision (p_total_after) can be calculated using the law of conservation of momentum as well.
p_total_after = p_ham_final + p_cart_final
Since the ham and the cart move together after the collision, they will have the same final velocity. Let's represent the final velocity with v_final.
p_total_after = (mass_ham + mass_cart) * v_final
p_total_after = (4.5 kg + 15 kg) * v_final
p_total_after = 19.5 kg * v_final
As p_total_before = p_total_after,
9.9 kg*m/s = 19.5 kg * v_final
Now we can calculate the final velocity (v_final):
v_final = 9.9 kg*m/s / 19.5 kg
v_final = 0.508 m/s
Therefore, the cart will travel at a speed of 0.508 m/s after the collision.
To find the duration of the collision (time), we need to use the force exerted during the collision and calculate the impulse.
Impulse (J) = force * time
Rearranging the formula, we get:
time = impulse / force
Given that the force (F) exerted during the collision is 2N, we can calculate the time of the collision:
time = J / F
However, before calculating the time, we need to determine the impulse (J). The impulse is equal to the change in momentum, which can be calculated as:
J = p_ham_final - p_ham_initial
Since the initial momentum of the ham is p_ham = 9.9 kg*m/s:
J = p_ham_final - 9.9 kg*m/s
To find p_ham_final, we can use the final velocity (v_final) and the mass of the ham (m_ham):
p_ham_final = m_ham * v_final
p_ham_final = 4.5 kg * 0.508 m/s
Substituting this value back into the equation for J:
J = 4.5 kg * 0.508 m/s - 9.9 kg*m/s
J = 2.286 kg*m/s
Now we can calculate the time:
time = J / F
time = 2.286 kg*m/s / 2 N
time = 1.143 s
Therefore, the collision between the ham and the cart lasted for approximately 1.143 seconds.
To determine the speed at which the cart will travel when the ham is thrown into it, we can apply the principle of conservation of momentum.
The momentum of an object is given by the product of its mass and velocity. According to the conservation of momentum, the total momentum before and after a collision remains constant, as long as no external forces act on the system.
The initial momentum of the ham can be calculated using the formula:
Initial momentum of ham = Mass of ham * Initial velocity of ham
P(ham) = m(ham) * v(ham)
= 4.5 kg * 2.2 m/s
= 9.9 kg*m/s
Since the shopping cart is initially stationary, its initial momentum is zero.
P(cart) = 0 kg*m/s
After the collision, the final momentum of the system (ham + cart) will be equal to the initial momentum.
P(final) = P(ham) + P(cart)
Since the cart and ham move together after the collision, we can find the velocity of the cart by rearranging the equation:
P(final) = (m(ham) + m(cart)) * v(cart)
v(cart) = P(final) / (m(ham) + m(cart))
Plugging in the values:
v(cart) = 9.9 kg*m/s / (4.5 kg + 15 kg)
= 9.9 kg*m/s / 19.5 kg
≈ 0.51 m/s
Therefore, the shopping cart will travel at approximately 0.51 m/s when the ham is thrown into it.
Now, to determine the duration of the collision, we need to consider the force applied by the ham during the collision.
Force can be defined as the rate of change of momentum:
Force = (Change in momentum) / (Time duration of the collision)
Rearranging the equation, we have:
Time duration of the collision = (Change in momentum) / Force
The change in momentum during the collision is equal to the final momentum of the system (ham + cart) minus the initial momentum of the system.
Change in momentum = P(final) - P(initial)
Plugging in the values, we can calculate the change in momentum:
Change in momentum = P(final) - P(initial)
= P(ham) + P(cart) - P(initial)
= P(ham) + P(cart) - [P(ham) + P(cart)] (since P(initial) = 0 kg*m/s)
= 0 kg*m/s
Therefore, the change in momentum during the collision is zero.
Since the force applied during the collision is 2 Newtons, and the change in momentum is zero, the duration of the collision can be calculated as:
Time duration of the collision = (Change in momentum) / Force
= 0 kg*m/s / 2 N
= 0 seconds
Hence, the collision between the ham and the cart lasts for an instantaneous moment, which means the duration of the collision is approximately zero seconds.
Use conservation of momentum for the first
question. They expect you to neglect friction. The ham will have to thrown in the direction the cart wheels are pointed.
4.5*2.2 = (4.5 + 15)*Vfinal
For the second question,
Force*Time = final cart momentum
= Vfinal*(15 kg)