In a very busy off-campus eatery one chef sends a 241-g broccoli-tomato-pickle-onion-mushroom pizza sliding down the counter from left to right at 1.53 m/s. Almost simultaneously, the other chef launches a 321-g veggieburger-on-a-bun (with everything, but hold the horseradish) along the same counter from right to left at 2.49 m/s. The two delicacies collide head on at the given speeds–the counter is practically friction-free due to accumulated grease–and merge into a single, unbelievably savory serving. At what speed does the delectible dish move?
The law of conservation of linear momentum for
inelastic collision
m1•v1 - m2•v2=(m1+m2)•u
u= {m1•v1 - m2•v2}/(m1+m2)=
={0.241•1.53 -0.321•2.49)/(0.241+0.321)=
=-0.766 m/s.
(in the direction of the second pizza motion)
To find the speed at which the merged pizza and veggieburger move, we can apply the principle of conservation of momentum.
The momentum of an object is given as the product of its mass and velocity. The principle of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.
Let's denote the momentum of the pizza before the collision as Pp, and the momentum of the veggieburger before the collision as Pb. After the collision, the merged dish will have a total momentum, which we'll denote as Pm.
The momentum of an object is calculated using the formula:
Momentum = mass * velocity
Given:
Mass of the pizza (mp) = 241 g = 0.241 kg
Velocity of the pizza (vp) = 1.53 m/s
Mass of the veggieburger (mb) = 321 g = 0.321 kg
Velocity of the veggieburger (vb) = -2.49 m/s (negative since it is moving in the opposite direction)
Now we can calculate the momenta before the collision:
Momentum of the pizza (Pp) = mp * vp = 0.241 kg * 1.53 m/s = 0.36933 kg.m/s
Momentum of the veggieburger (Pb) = mb * vb = 0.321 kg * -2.49 m/s = -0.79879 kg.m/s
Next, we can add the two momenta together to find the total momentum after the collision:
Total momentum after the collision (Pm) = Pp + Pb
Pm = 0.36933 kg.m/s + (-0.79879 kg.m/s)
Pm = -0.42946 kg.m/s
Finally, to find the velocity of the merged dish, we divide the total momentum by the total mass (which is the sum of the individual masses of pizza and veggieburger):
Total mass (m) = mp + mb = 0.241 kg + 0.321 kg = 0.562 kg
Velocity (v) = Pm / m
v = -0.42946 kg.m/s / 0.562 kg
v ≈ -0.764 m/s
Therefore, the merged dish moves at approximately -0.764 m/s. The negative sign indicates that it is moving in the opposite direction from the initial motion of the veggieburger.
To find the speed at which the combined dish moves after the collision, we need to apply the principle of conservation of momentum.
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, the system consists of the broccoli-tomato-pickle-onion-mushroom pizza and the veggieburger-on-a-bun.
The momentum of an object is defined as the product of its mass and velocity. Therefore, we can calculate the initial momentum of each object before the collision as follows:
Pizza momentum = mass of pizza * velocity of pizza
= 241 g * 1.53 m/s
Burger momentum = mass of burger * velocity of burger
= 321 g * (-2.49 m/s) (since it is moving in the opposite direction)
Now, since the pizza and burger collide and stick together, they form a single object after the collision. Let's call the final combined mass mf and the final combined velocity vf.
Using the principle of conservation of momentum, the total initial momentum must equal the total final momentum:
Total initial momentum = Total final momentum
(Pizza momentum) + (Burger momentum) = (Combined dish momentum)
(241 g * 1.53 m/s) + (321 g * (-2.49 m/s)) = (mf * vf)
Solving this equation will give us the mass of the combined dish and its final velocity.
(mf * vf) = (361.41 g*m/s - 800.79 g*m/s)
Since the masses are in grams and the velocities are in m/s, the units of momentum are gram-meters per second (g*m/s).
The combined dish's final mass is the sum of the individual masses:
mf = (mass of pizza) + (mass of burger)
= 241 g + 321 g
Finally, we can calculate the final velocity of the combined dish by dividing the total momentum by its mass:
vf = (combined dish momentum) / mf
Substituting the values, we can solve for vf.