1) Find the final velocity of the two balls if the ball with velocity
v2i = −21.0 cm/s has a mass equal to half that of the ball with initial velocity
v1i = +26.7 cm/s. (Indicate the direction with the sign of your answer.)
*just confused on how to solve for V1f / V2f=42cm/s*
2) A 2.62 kg object initially moving in the positive x-direction with a velocity of +5.43 m/s collides with and sticks to a 1.98 kg object initially moving in the negative y-direction with a velocity of -3.03 m/s. Find the final components of velocity of the composite object. (Indicate the direction with the sign of your answer.) ( vfx = m/s vfy = m/s)
3) A 1,275-kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a 9,400-kg truck moving in the same direction at 20.0 m/s The velocity of the car right after the collision is 18.0 m/s to the east. (I solved for part a, which states what is the velocity of the truck right after the collision? which equals (20.9495m/s east), confused on how to answer part(b). The figure numbers shows before +25.0m/s plus +20.0m/s which yields after that is +18.0m/s plus part a answer (20.9495m/s east)
(b) How much mechanical energy is lost in the collision?
please any help would be wonderful
first off, since V1f / V2f are both speeds, their ratio cannot be in m/s
It will just be a number.
All of these problems are the same, and involve conservation of momentum. In each case, you need
m1v1 + m1v1 = m2v2 + m2v2
By the time you plug in the given numbers, you can solve for the unknown value. Be sure to be careful with the signs on the velocities.
1) To find the final velocity of the two balls, we can use the conservation of momentum principle. Since momentum is conserved, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the final velocities of the first ball as v1f and the second ball as v2f.
The initial momentum is calculated as:
m1 * v1i + m2 * v2i
The final momentum is calculated as:
m1 * v1f + m2 * v2f
We have the following information:
m2 = 0.5 * m1 (since the mass of the second ball is half that of the first ball)
v1i = +26.7 cm/s
v2i = -21.0 cm/s
v1f = ?
v2f = 42 cm/s
Using the conservation of momentum principle, we can set up the equation:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Substituting the given values, we get:
m1 * 26.7 cm/s + 0.5 * m1 * (-21.0 cm/s) = m1 * v1f + 0.5 * m1 * 42 cm/s
Simplifying the equation, we get:
26.7 - 0.5 * 21.0 = v1f + 0.5 * 42
5.7 = v1f + 21.0
v1f = 5.7 - 21.0
v1f = -15.3 cm/s
Therefore, the final velocity of the first ball is -15.3 cm/s.
Since the final velocity of the second ball is given as 42 cm/s, we can state the final velocities as:
v1f = -15.3 cm/s (negative sign indicating opposite direction)
v2f = 42 cm/s
2) To find the final components of velocity of the composite object, we can again use the conservation of momentum principle. Since momentum is conserved, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the final components of velocity as vfx and vfy.
The initial momentum is calculated as:
m1 * v1i + m2 * v2i
The final momentum is calculated as:
(m1 + m2) * vfx + (m1 + m2) * vfy
We have the following information:
m1 = 2.62 kg
m2 = 1.98 kg
v1i = +5.43 m/s
v2i = -3.03 m/s
vfx = ?
vfy = ?
Using the conservation of momentum principle, we can set up the equation:
m1 * v1i + m2 * v2i = (m1 + m2) * vfx + (m1 + m2) * vfy
Substituting the given values, we get:
2.62 kg * 5.43 m/s + 1.98 kg * (-3.03 m/s) = (2.62 kg + 1.98 kg) * vfx + (2.62 kg + 1.98 kg) * vfy
Simplifying the equation, we get:
14.2186 - 6.0294 = 4.6 kg * vfx + 4.6 kg * vfy
8.1892 = 4.6 * (vfx + vfy)
Since the composite object sticks together after the collision, its final velocities in the x and y directions will be the same. Therefore, we can denote both final velocities as vf.
8.1892 = 9.4 * vf
Solving for vf:
vf = 8.1892 / 9.4 m/s
vf ≈ 0.872 m/s
Therefore, the final components of velocity of the composite object are:
vfx = 0.872 m/s (positive sign indicating the initial positive x-direction)
vfy = 0.872 m/s (positive sign indicating the initial negative y-direction)
3) To answer part (b), we need to calculate the amount of mechanical energy lost in the collision.
Mechanical energy is conserved only in completely elastic collisions. In a collision where objects stick together, like in this case, some mechanical energy is lost in the form of sound, heat, and deformation.
To calculate the mechanical energy lost, we need to compare the initial mechanical energy (before the collision) with the final mechanical energy (after the collision).
The initial mechanical energy is given by:
Initial kinetic energy of the car + Initial kinetic energy of the truck
= 0.5 * m_car * v_car^2 + 0.5 * m_truck * v_truck^2
The final mechanical energy is given by:
Final kinetic energy of the car + Final kinetic energy of the truck
= 0.5 * m_car * v_car_final^2 + 0.5 * m_truck * v_truck_final^2
We have the following information:
m_car = 1,275 kg
m_truck = 9,400 kg
v_car = 25.0 m/s
v_truck = 20.0 m/s
v_car_final = 18.0 m/s
v_truck_final = 20.9495 m/s (from part a)
Using the formulas above, we can calculate the initial and final mechanical energies and find the difference:
Initial mechanical energy = 0.5 * 1,275 kg * (25.0 m/s)^2 + 0.5 * 9,400 kg * (20.0 m/s)^2
Final mechanical energy = 0.5 * 1,275 kg * (18.0 m/s)^2 + 0.5 * 9,400 kg * (20.9495 m/s)^2
Mechanical energy lost = Initial mechanical energy - Final mechanical energy
Simply substitute the values into the equations and calculate to find the mechanical energy lost in the collision.
1) To solve for the final velocities of the two balls, you can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The initial momentum of the system is given by:
P_initial = m1 * v1i + m2 * v2i
The final momentum of the system is given by:
P_final = m1 * v1f + m2 * v2f
Since the balls stick together after the collision, their final velocities are equal:
v1f = v2f = vf
Using the conservation of momentum, we can set the initial and final momenta equal to each other:
m1 * v1i + m2 * v2i = (m1 + m2) * vf
Now substitute the given values into the equation and solve for vf:
(0.5 * m1) * v1i + m2 * v2i = (0.5 * m1 + m2) * vf
(0.5 * 26.7 cm/s) + (-21.0 cm/s) = (0.5 * 26.7 cm/s + 2 * 21.0 cm/s) * vf
13.35 cm/s - 21.0 cm/s = (13.35 cm/s + 42.0 cm/s) * vf
-7.65 cm/s = 55.35 cm/s * vf
Dividing both sides by 55.35 cm/s:
vf = -7.65 cm/s / 55.35 cm/s
vf ≈ -0.1387 cm/s
Therefore, the final velocity of both balls is approximately -0.1387 cm/s.
2) To find the final components of velocity of the composite object after the collision, you can again use the principle of conservation of momentum.
The initial momentum of the system is given by:
P_initial = m1 * v1i + m2 * v2i
The final momentum of the system is given by:
P_final = (m1 + m2) * vf
Since the objects stick together after the collision, their final velocities are equal in magnitude and opposite in direction.
Let's assume the final velocity of the composite object is (vfx, vfy).
Then we have two equations based on the conservation of momentum:
m1 * v1i + m2 * v2i = (m1 + m2) * vfx
m1 * v1i + m2 * v2i = (m1 + m2) * vfy
Substituting the given values:
(2.62 kg) * (5.43 m/s) + (1.98 kg) * (-3.03 m/s) = (2.62 kg + 1.98 kg) * vfx
(2.62 kg) * (5.43 m/s) + (1.98 kg) * (-3.03 m/s) = (2.62 kg + 1.98 kg) * vfy
Solving these equations:
vfx ≈ (2.62 kg * 5.43 m/s + 1.98 kg * -3.03 m/s) / (2.62 kg + 1.98 kg)
vfy ≈ (2.62 kg * 5.43 m/s + 1.98 kg * -3.03 m/s) / (2.62 kg + 1.98 kg)
Calculate the values to find the final components of velocity of the composite object.
3) (b) To calculate the mechanical energy lost in the collision, you can subtract the final kinetic energy of the system after the collision from the initial kinetic energy of the system before the collision.
The initial kinetic energy of the system is given by:
KE_initial = 1/2 * m1 * v1i^2 + 1/2 * m2 * v2i^2
The final kinetic energy of the system is given by:
KE_final = 1/2 * (m1 + m2) * vf^2
The mechanical energy lost in the collision is:
Energy_lost = KE_initial - KE_final
Substitute the given values into the equations and calculate the mechanical energy lost in the collision.
1) To solve for the final velocities of the two balls, you can use the principle of conservation of momentum. In this case, the initial momentum of the system is equal to the final momentum.
The equation for momentum is given by:
p = mv
First, let's calculate the initial momentum of the system. For the first ball with mass m1 and initial velocity v1i:
p1i = m1 * v1i
For the second ball with mass m2 and initial velocity v2i:
p2i = m2 * v2i
To find the final velocities v1f and v2f, we set up the equation:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Since we are given the masses and initial velocities, we can substitute the values into the equation and solve for the final velocities.
2) In this problem, we can again use the principle of conservation of momentum to solve for the final velocities of the composite object.
The equation for momentum is given by:
p = mv
Let's consider the x-component and y-component separately.
In the x-component:
For the first object with mass m1 and initial velocity v1i:
p1i_x = m1 * v1i_x
For the second object with mass m2 and initial velocity v2i:
p2i_x = m2 * v2i_x
To find the final x-component velocity vfx, we set up the equation:
m1 * v1i_x + m2 * v2i_x = (m1 + m2) * vfx
Similarly, in the y-component:
For the first object with mass m1 and initial velocity v1i:
p1i_y = m1 * v1i_y
For the second object with mass m2 and initial velocity v2i:
p2i_y = m2 * v2i_y
To find the final y-component velocity vfy, we set up the equation:
m1 * v1i_y + m2 * v2i_y = (m1 + m2) * vfy
Substituting the given masses and initial velocities, you can solve for the final velocities in the x and y directions.
3) To solve part (b) of the problem, we need to find the amount of mechanical energy lost in the collision.
Mechanical energy is given by the sum of kinetic energy and potential energy. In this case, since we are dealing with objects moving horizontally, we can ignore any changes in potential energy.
The equation for kinetic energy is given by:
KE = (1/2) * m * v^2
To find the initial kinetic energy of the system, we can calculate the kinetic energy of each object before the collision and sum them up.
For the car:
KE1i = (1/2) * m1 * v1i^2
For the truck:
KE2i = (1/2) * m2 * v2i^2
The total initial kinetic energy is given by:
KEi = KE1i + KE2i
After the collision, the final kinetic energy of the system is the sum of the kinetic energies of the car and the truck.
For the car:
KE1f = (1/2) * m1 * v1f^2
For the truck:
KE2f = (1/2) * m2 * v2f^2
The total final kinetic energy is given by:
KEf = KE1f + KE2f
The amount of mechanical energy lost in the collision is then:
Energy lost = KEi - KEf
Substituting the given masses and velocities, you can calculate the mechanical energy lost.