A(n) 19 g object moving to the right at
39 cm/s overtakes and collides elastically with
a 36 g object moving in the same direction at
19 cm/s.
Find the velocity of faster object after the
collision.
Answer in units of cm/s.
(I know this is a lot similar to another problem that shows how to find the velocity of the slower object, but when finding the velocity of the faster object, do you solve the same way, if not, then how do you solve to get the velocity of the faster object?)
Given: M1 = 19 g, V1 = 39 cm/s.
M2 = 36 g, V2 = 19 cm/s.
V3 = velocity of M1 after collision.
V3 = ((M1-M2)V1 + 2M2*V2)/(M1+M2)
V3 = (-17*39 + 72*19)/55 = 705/55 = 12.8 cm/s.
To find the velocity of the faster object after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's denote the initial velocity of the 19g object as v1 and the initial velocity of the 36g object as v2. According to the problem, v1 = 39 cm/s and v2 = 19 cm/s.
Using conservation of momentum, we have:
Initial momentum = Final momentum
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Where:
m1 = mass of the 19g object = 19g = 0.019kg
v1 = initial velocity of the 19g object = 39 cm/s
m2 = mass of the 36g object = 36g = 0.036kg
v2 = initial velocity of the 36g object = 19 cm/s
v1' = final velocity of the 19g object (which we need to find)
v2' = final velocity of the 36g object (which we don't need for this question)
Substituting the given values, we have:
(0.019 * 39) + (0.036 * 19) = (0.019 * v1') + (0.036 * v2')
Simplifying this equation gives:
0.741 + 0.684 = 0.019v1' + 0.036v2'
Now, using conservation of kinetic energy, we have:
Initial kinetic energy = Final kinetic energy
(1/2 * m1 * (v1)^2) + (1/2 * m2 * (v2)^2) = (1/2 * m1 * (v1')^2) + (1/2 * m2 * (v2')^2)
Substituting the given values, we have:
(1/2 * 0.019 * (39)^2) + (1/2 * 0.036 * (19)^2) = (1/2 * 0.019 * (v1')^2) + (1/2 * 0.036 * (v2')^2)
Simplifying this equation gives:
14.265 + 6.228 = 0.0095(v1')^2 + 0.018(v2')^2
Now we have two equations:
0.741 + 0.684 = 0.019v1' + 0.036v2'
14.265 + 6.228 = 0.0095(v1')^2 + 0.018(v2')^2
We can solve these equations simultaneously to find the values of v1' and v2'. However, in this problem, we only need the value of v1' (velocity of the faster object).
Solving the first equation for v2', we get:
v2' = (0.741 + 0.684 - 0.019v1') / 0.036
Substituting this value of v2' into the second equation gives:
14.265 + 6.228 = 0.0095(v1')^2 + 0.018((0.741 + 0.684 - 0.019v1') / 0.036)^2
Now, we can solve this quadratic equation to find the value of v1'.
To solve this problem, we need to apply the principles of conservation of momentum and kinetic energy.
Step 1: Write down the given information.
Mass of the first object (faster object) = 19 g = 0.019 kg
Initial velocity of the first object = 39 cm/s
Mass of the second object (slower object) = 36 g = 0.036 kg
Initial velocity of the second object = 19 cm/s
Step 2: Calculate the initial momentum of each object.
Momentum = mass × velocity
Momentum of the first object = (0.019 kg) × (39 cm/s)
Momentum of the second object = (0.036 kg) × (19 cm/s)
Step 3: Apply the principle of conservation of momentum before the collision.
Total initial momentum = Total final momentum
(momentum of the first object before collision) + (momentum of the second object before collision) = (momentum of the first object after collision) + (momentum of the second object after collision)
Step 4: Solve the equation to find the velocity of the faster object after the collision.
(0.019 kg) × (39 cm/s) + (0.036 kg) × (19 cm/s) = (0.019 kg) × (V1) + (0.036 kg) × (V2)
Here, V1 and V2 represent the velocities of the first and second objects, respectively, after the collision.
Step 5: Use the conservation of kinetic energy to solve for V1 and V2.
The kinetic energy before the collision is equal to the kinetic energy after the collision because the collision is elastic.
The kinetic energy of an object is given by the equation: Kinetic energy = (1/2) × mass × velocity^2.
So, we have:
(1/2) × (0.019 kg) × (39 cm/s)^2 + (1/2) × (0.036 kg) × (19 cm/s)^2 = (1/2) × (0.019 kg) × (V1)^2 + (1/2) × (0.036 kg) × (V2)^2
Step 6: Solve the equation obtained in Step 4 and Step 5 simultaneously to find the values of V1 and V2.
By solving these equations, you will find the velocities of both objects after the collision. The velocity of the faster object (V1) can be determined by substituting the values of V2 and other known variables into the equation obtained in Step 4.
Note: The approach is similar for finding the velocity of the slower object, but the correct values will be substituted into different equations.