To solve this problem, we need to use the principles of conservation of momentum and kinetic energy because the collision is elastic.

The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:

m1*u1 + m2*u2 = m1*v1 + m2*v2

where:
m1 and m2 are the masses of the objects,
u1 and u2 are the initial velocities of the objects,
v1 and v2 are the final velocities of the objects.

Given that the 30-kg object (m1) is initially at rest, u1 = 0. The 10-kg object (m2) is initially moving at +3.0 m/s, so u2 = +3.0 m/s. We are asked to find v1, the final velocity of the 30-kg object.

Substituting the given values into the momentum equation, we get:

30*0 + 10*3 = 30*v1 + 10*v2
= 30 kg*m/s = 30*v1 + 10*v2 ----(1)

Since the collision is elastic, the kinetic energy is also conserved. The conservation of kinetic energy can be expressed as:

0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2

Substituting the given values into the kinetic energy equation, we get:

0.5*30*0^2 + 0.5*10*3^2 = 0.5*30*v1^2 + 0.5*10*v2^2
= 45 J = 15*v1^2 + 5*v2^2 ----(2)

We now have two equations (1 and 2) with two unknowns (v1 and v2). We can solve these equations simultaneously to find the values of v1 and v2.

However, the question only asks for the final velocity of the 30 kg object (v1), so we don't need to solve for v2. We can rearrange equation (1) to express v2 in terms of v1:

v2 = (30 - 30*v1) / 10
= v2 = 3 - 3*v1

Substitute v2 into equation (2):

45 = 15*v1^2 + 5*(3 - 3*v1)^2
= 45 = 15*v1^2 + 45 - 30*v1 + 45*v1^2
= 90*v1^2 - 30*v1 - 45 = 0

Divide the entire equation by 30:

3*v1^2 - v1 - 1.5 = 0

Solving this quadratic equation for v1, we get two solutions: v1 = 1 and v1 = -0.5. However, since the velocity cannot be negative in this context, we discard the -0.5 solution.

Therefore, the final velocity of the 30 kg object after the collision is +1.0 m/s. So, the correct answer is c.−1.0 m/s.

Question

To solve this problem, we need to use the principles of conservation of momentum and kinetic energy because the collision is elastic.

The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:

m1*u1 + m2*u2 = m1*v1 + m2*v2

where:
m1 and m2 are the masses of the objects,
u1 and u2 are the initial velocities of the objects,
v1 and v2 are the final velocities of the objects.

Given that the 30-kg object (m1) is initially at rest, u1 = 0. The 10-kg object (m2) is initially moving at +3.0 m/s, so u2 = +3.0 m/s. We are asked to find v1, the final velocity of the 30-kg object.

Substituting the given values into the momentum equation, we get:

30*0 + 10*3 = 30*v1 + 10*v2
=> 30 kg*m/s = 30*v1 + 10*v2 ----(1)

Since the collision is elastic, the kinetic energy is also conserved. The conservation of kinetic energy can be expressed as:

0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2

Substituting the given values into the kinetic energy equation, we get:

0.5*30*0^2 + 0.5*10*3^2 = 0.5*30*v1^2 + 0.5*10*v2^2
=> 45 J = 15*v1^2 + 5*v2^2 ----(2)

We now have two equations (1 and 2) with two unknowns (v1 and v2). We can solve these equations simultaneously to find the values of v1 and v2.

However, the question only asks for the final velocity of the 30 kg object (v1), so we don't need to solve for v2. We can rearrange equation (1) to express v2 in terms of v1:

v2 = (30 - 30*v1) / 10
=> v2 = 3 - 3*v1

Substitute v2 into equation (2):

45 = 15*v1^2 + 5*(3 - 3*v1)^2
=> 45 = 15*v1^2 + 45 - 30*v1 + 45*v1^2
=> 90*v1^2 - 30*v1 - 45 = 0

Divide the entire equation by 30:

3*v1^2 - v1 - 1.5 = 0

Solving this quadratic equation for v1, we get two solutions: v1 = 1 and v1 = -0.5. However, since the velocity cannot be negative in this context, we discard the -0.5 solution.

Therefore, the final velocity of the 30 kg object after the collision is +1.0 m/s. So, the correct answer is c.−1.0 m/s.

Knowee AI · Accepted Answer