a 10.0-g marble rolls to the left with a velocity of magnitude 0.400 m.s^-1 on a smooth, level surface and makes a head-on collision with a larger 30.0-g marble rolling to right with a velocity of magnitude 0.100 m.s^-1. If the collision is perfectly elastic,find the velocity of each marble after the collision.( since the collision is head-on, all the motion os along a line.)
v1 – to the left,
v2 – to the right
u1=[(m1-m2) •v1 – 2• m2•v2]/(m1+m2) =
=[(10-30) •0.4 - 2•30•0.1]/(10+30) = - 0.35 m/s.
u2 = [2•m1•v1 –(m2-m1) •v2]/(m1+m2) =
= [2•10•0.4 –(30-10) •0.1]/(10+30) =0.15 m/s.
u1 – to the right,
u2 – to the left.
To solve this problem, we can use the principle of conservation of momentum and kinetic energy in an elastic collision.
Step 1: Calculate the initial momentum of each marble:
The momentum of an object is given by the product of its mass and velocity. Therefore, we can calculate the initial momentum of the left marble (marble 1) and the right marble (marble 2) as follows:
Initial momentum of marble 1 (P1) = mass of marble 1 (m1) × velocity of marble 1 (v1)
= 10.0 g × 0.400 m/s
Initial momentum of marble 2 (P2) = mass of marble 2 (m2) × velocity of marble 2 (v2)
= 30.0 g × (-0.100 m/s) [negative direction indicates the opposite motion]
Note that the negative sign in front of the velocity of marble 2 accounts for its opposing direction of motion.
Step 2: Calculate the total initial momentum before the collision:
Since the collision is head-on, the total initial momentum of both marbles is equal to the sum of their individual momenta:
Initial total momentum (P) = P1 + P2
Step 3: Apply the principle of conservation of momentum:
According to the law of conservation of momentum, the total momentum of the system (both marbles) should remain constant before and after the collision. Therefore, after the collision, the total momentum will still be equal to P.
Step 4: Calculate the final velocity of each marble after the collision:
Let's assume that after the collision, marble 1 moves with a velocity v1' and marble 2 moves with a velocity v2'.
Final momentum of marble 1 (P1') = mass of marble 1 (m1) × velocity of marble 1 (v1')
Final momentum of marble 2 (P2') = mass of marble 2 (m2) × velocity of marble 2 (v2')
Applying the principle of conservation of momentum:
P = P1' + P2'
Step 5: Solve for the final velocities:
Now we can substitute the values and solve for the final velocities of each marble.
10.0 g × 0.400 m/s + 30.0 g × (-0.100 m/s) = 10.0 g × v1' + 30.0 g × v2'
After simplifying and dividing by the respective masses:
0.004 kg · m/s + (-0.003 kg · m/s) = 0.01 kg · v1' + 0.03 kg · v2'
0.001 kg · m/s = 0.01 kg · v1' + 0.03 kg · v2'
From this equation, we can see that the final velocities of the marbles depend on each other since they move in opposite directions.
Please let me know if you need assistance with further calculations or if you have any additional questions.
To solve this problem, we can apply the principle of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision. The momentum of an object is calculated by multiplying its mass by its velocity.
Let's denote the velocity of the smaller marble before the collision as v1, and the velocity of the larger marble as v2. The mass of the smaller marble is 10.0 g (or 0.010 kg), and the mass of the larger marble is 30.0 g (or 0.030 kg).
According to the principle of conservation of momentum, we have:
Initial momentum = Final momentum
For the smaller marble:
Initial momentum = mass of smaller marble × velocity of smaller marble before collision
= 0.010 kg × 0.400 m/s
= 0.004 kg⋅m/s
For the larger marble:
Initial momentum = mass of larger marble × velocity of larger marble before collision
= 0.030 kg × (-0.100 m/s) (since the velocity is towards the right)
= -0.003 kg⋅m/s
Now, let's consider the final velocities of the marbles after the collision. Since the collision is perfectly elastic, the total kinetic energy is conserved. This means that the sum of the kinetic energies of the marbles before the collision is equal to the sum of the kinetic energies after the collision.
The kinetic energy of an object is calculated by multiplying half the mass by the square of the velocity.
For the smaller marble:
Initial kinetic energy = (1/2) × mass of smaller marble × (velocity of smaller marble before collision)^2
= 0.5 × 0.010 kg × (0.400 m/s)^2
= 0.004 J
For the larger marble:
Initial kinetic energy = (1/2) × mass of larger marble × (velocity of larger marble before collision)^2
= 0.5 × 0.030 kg × (-0.100 m/s)^2
= 0.00015 J
Since the total kinetic energy is conserved, we have:
Initial kinetic energy = Final kinetic energy
Final kinetic energy = (1/2) × mass of smaller marble × (velocity of smaller marble after collision)^2
+ (1/2) × mass of larger marble × (velocity of larger marble after collision)^2
Let's assume the velocity of the smaller marble after the collision is v1' and the velocity of the larger marble after the collision is v2'.
Therefore, we can write the equation as:
0.004 J + 0.00015 J = (1/2) × 0.010 kg × (v1')^2 + (1/2) × 0.030 kg × (v2')^2
Simplifying the equation:
0.00415 J = 0.005 kg × (v1')^2 + 0.015 kg × (v2')^2
Now, we can use the principle of conservation of momentum to express v1' and v2' in terms of the initial velocities:
0.004 kg⋅m/s = 0.010 kg × v1' + 0.030 kg × v2' (Equation 1)
Solving Equations 1 and 2 simultaneously will give us the values of v1' and v2', which are the final velocities of the marbles after the collision.