Question

Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing blow. Asteroid A, which was initially traveling at 40.0 m/s, is deflected 30 degrees from its original direction, while asteroid B, which was initially at rest, travels at 45 degrees to the original direction of A.

(a) Find the speed of each asteroid after the collision.
(b) What fraction of the original kinetic energy of asteroid A dissipates during this collision?

Answer 1

When two asteroids collide with a glancing blow, momentum and kinetic energy are conserved. To find the speed of each asteroid after the collision, use the equation for conservation of momentum. To find the fraction of the original kinetic energy dissipated, use the equation for kinetic energy.

Step-by-step explanation:

When two asteroids collide with a glancing blow, momentum and kinetic energy are conserved.

(a) To find the speed of each asteroid after the collision, we can use the conservation of momentum.

Let's assume asteroid A's final speed is v1 and asteroid B's final speed is v2.

Using the equation for conservation of momentum, we get:

m1 * v1 = m1 * v1' + m2 * v2'

where m1 and m2 are the masses of asteroids A and B, respectively, and v1' and v2' are their initial velocities.

Since asteroid B was initially at rest, v2' is 0.

Also, the initial velocity of asteroid A is given (40.0 m/s).

The final velocities of asteroid A and B can be found by solving these equations simultaneously.

(b) To find the fraction of the original kinetic energy dissipated, we can use the equation for kinetic energy:

K = 0.5 * m * v^2

where K is the kinetic energy and m and v are the mass and velocity of the asteroid, respectively.

We can calculate the initial kinetic energy of asteroid A and the final kinetic energy of both asteroids after the collision, then find the ratio of the dissipated energy to the initial energy.

Answer 2

To find the speed of each asteroid after the collision, we can use the laws of conservation of momentum and conservation of kinetic energy. To find the fraction of the original kinetic energy dissipated, we can compare the initial and final kinetic energies.

Step-by-step explanation:

To solve this question, we can start by applying the laws of conservation of momentum and conservation of kinetic energy. Using the given information, we can calculate the final velocities of both asteroids after the collision.

Calculating the final velocities:

Let asteroid A's final velocity be Va and asteroid B's final velocity be Vb.

Using the conservation of momentum, we know that the total momentum before the collision is equal to the total momentum after the collision.

Setting up momentum equations for both asteroids:

Asteroid A: (mass of A) * (initial velocity of A) = (mass of A) * (final velocity of A)

Asteroid B: (mass of B) * (initial velocity of B) = (mass of B) * (final velocity of B)

Substituting the given values:

(15 x 10³ kg) * (40.0 m/s) = (15 x 10³ kg) * Va

(20 x 10³ kg) * 0 = (20 x 10³ kg) * Vb

We can solve these equations to find Va and Vb.

Calculating the fraction of the original kinetic energy dissipated:

The initial kinetic energy of asteroid A is given by: (1/2) * (mass of A) * (initial velocity of A)^2.

The final kinetic energy of asteroid A is given by: (1/2) * (mass of A) * (final velocity of A)^2.

The fraction of kinetic energy dissipated is: (initial kinetic energy of A - final kinetic energy of A) / initial kinetic energy of A.

Substituting the given values, we can calculate the fraction.