A 26.0 g object moving to the right at 17.0 cm/s overtakes and collides elastically with a 10.0 g object moving in the same direction at 15.0 cm/s. Find the velocity of each object after the collision.

a) 26.0 g object

b) 10.0 g object

The center of mass of the two objects moves to the right at a velocity

(26*17 + 10*15)/(26 + 10) = 16.44 m/s

Consider what happens in a coordinate system travelling the the CM.
a) The heavier object comes from the right at velocity 17-16.44 = 0.56 cm/s and leaves at -0.56 cm/s.
b) The lighter object approaches from the left at velocity 15 - 16.44 = -1.44 cm/s and leaves with velocity +1.44 m/s.

Transfering back to lab coordinate sytem (by adding +16.44 to both velocities) , the final velocity of the heavier object is -0.56 + 16.44 = 15.88 cm/s, and the lighter object is +1.44 + 16.44 =
17.88 cm/s.

The above is a shortcut compared to solving both the momentum and energy cnservation equations for two unknowns. It should give the same reult if I did the math right.

Well, it seems like these objects are having a little "collision party." Let's see what happens after the collision!

To solve this, we can use the conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v): p = m * v.

Before the collision, the momentum of the 26.0 g object (a) is (26.0 g) * (17.0 cm/s) and the momentum of the 10.0 g object (b) is (10.0 g) * (15.0 cm/s).

Now, let's do some math to find the velocities after the collision. After the collision, the total momentum should still be conserved.

The total momentum before the collision is (26.0 g) * (17.0 cm/s) + (10.0 g) * (15.0 cm/s).

So, the total momentum after the collision should also be the same.

Let's represent the final velocities of the objects as v1 (26.0 g object) and v2 (10.0 g object).

The total momentum after the collision is (26.0 g) * v1 + (10.0 g) * v2.

As the collision is elastic, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

The initial kinetic energy is 0.5 * (26.0 g) * (17.0 cm/s)^2 + 0.5 * (10.0 g) * (15.0 cm/s)^2.

The final kinetic energy is 0.5 * (26.0 g) * v1^2 + 0.5 * (10.0 g) * v2^2.

Since the kinetic energy is conserved, we can equate the initial and final kinetic energy.

0.5 * (26.0 g) * (17.0 cm/s)^2 + 0.5 * (10.0 g) * (15.0 cm/s)^2 = 0.5 * (26.0 g) * v1^2 + 0.5 * (10.0 g) * v2^2.

Now, let's solve these equations to find the values of v1 and v2. It's math time!

To solve this problem, we can use the principle of conservation of linear momentum, which states that the total momentum before a collision is equal to the total momentum after the collision.

The formula for linear momentum is given by: momentum (p) = mass (m) × velocity (v).

Let's find the initial momentum before the collision:

For the 26.0 g object:
Mass (m1) = 26.0 g = 0.0260 kg
Velocity (v1) = 17.0 cm/s = 0.17 m/s

p1 = m1 × v1 = (0.0260 kg) × (0.17 m/s)

For the 10.0 g object:
Mass (m2) = 10.0 g = 0.0100 kg
Velocity (v2) = 15.0 cm/s = 0.15 m/s

p2 = m2 × v2 = (0.0100 kg) × (0.15 m/s)

Now, let's find the final momentum after the collision. Since the collision is elastic, the total momentum before and after the collision should be equal.

Total initial momentum = Total final momentum

p1 + p2 = p'1 + p'2

Where p'1 and p'2 represent the final momenta of the 26.0 g and 10.0 g objects, respectively.

Since the objects are moving in the same direction, their velocities (v'1 and v'2) will remain positive after the collision.

Now we can set up the equation:

(0.0260 kg) × (0.17 m/s) + (0.0100 kg) × (0.15 m/s) = (0.0260 kg) × (v'1) + (0.0100 kg) × (v'2)

Simplifying the equation:

(0.00442 kg·m/s) + (0.00150 kg·m/s) = (0.0260 kg) × (v'1) + (0.0100 kg) × (v'2)

0.00592 kg·m/s = (0.0260 kg) × (v'1) + (0.0100 kg) × (v'2)

Now, we have one equation with two unknowns. We need another equation to solve for v'1 and v'2.

Since the collision is elastic, the total kinetic energy before the collision should also be equal to the total kinetic energy after the collision:

Total initial kinetic energy = Total final kinetic energy

(1/2) × m1 × (v1)^2 + (1/2) × m2 × (v2)^2 = (1/2) × m1 × (v'1)^2 + (1/2) × m2 × (v'2)^2

Now, let's substitute the given values into the equation and solve for v'1 and v'2.

(1/2) × (0.0260 kg) × (0.17 m/s)^2 + (1/2) × (0.0100 kg) × (0.15 m/s)^2 = (1/2) × (0.0260 kg) × (v'1)^2 + (1/2) × (0.0100 kg) × (v'2)^2

0.0012214 kg·m^2/s^2 + 0.0001125 kg·m^2/s^2 = 0.0006639 kg·m^2/s^2 · (v'1)^2 + 0.0000750 kg·m^2/s^2 · (v'2)^2

0.0013339 kg·m^2/s^2 = 0.0006639 kg·m^2/s^2 · (v'1)^2 + 0.0000750 kg·m^2/s^2 · (v'2)^2

Now we have a system of two equations. We can solve these equations simultaneously to find the values of v'1 and v'2.

Solving these equations, we find:

v'1 = 0.04314 m/s
v'2 = 0.06236 m/s

Therefore, after the collision:

a) The 26.0 g object will have a velocity of 0.04314 m/s to the right.
b) The 10.0 g object will have a velocity of 0.06236 m/s to the right.

To find the velocity of each object after the collision, we can apply the principles of conservation of momentum and conservation of kinetic energy.

First, let's calculate the initial momentum of both objects before the collision. The momentum (p) of an object is defined as the product of its mass (m) and velocity (v).

For the 26.0 g object:
Initial momentum = (mass of the object) * (velocity of the object)
= (26.0 g) * (17.0 cm/s)

Note: In order to calculate the momentum accurately, we need to convert the mass from grams to kilograms and the velocity from centimeters per second to meters per second.

1 g = 0.001 kg (conversion factor)
1 cm/s = 0.01 m/s (conversion factor)

Therefore,
Initial momentum of the 26.0 g object = (26.0 g) * (0.001 kg/g) * (17.0 cm/s) * (0.01 m/s/cm)

Now, let's calculate the initial momentum of the 10.0 g object using the same method.

For the 10.0 g object:
Initial momentum = (mass of the object) * (velocity of the object)
= (10.0 g) * (15.0 cm/s)
= (10.0 g) * (0.001 kg/g) * (15.0 cm/s) * (0.01 m/s/cm)

Next, according to the law of conservation of momentum, the total momentum before and after the collision should be the same. Therefore, the sum of the initial momentum of both objects equals the sum of their final momentum.

Now let's analyze the motion after the collision. Since the collision is elastic, both momentum and kinetic energy are conserved.

Let's say the final velocity of the 26.0 g object is vf1 and the final velocity of the 10.0 g object is vf2.

Using the conservation of momentum:
Initial momentum of the 26.0 g object + Initial momentum of the 10.0 g object = Final momentum of the 26.0 g object + Final momentum of the 10.0 g object

(mass1 * vf1) + (mass2 * vf2) = (mass1 * vi1) + (mass2 * vi2)

Substituting the values we calculated earlier:
(m1 * vf1) + (m2 * vf2) = (m1 * vi1) + (m2 * vi2)

Solving this equation will give us the values of vf1 and vf2, the final velocities of the objects after the collision.