A 0.400 glider is moving to the right on a frictionless, horizontal air track with a speed of 0.800 when it makes a head-on collision with a stationary 0.150 glider.Find the magnitude of the final velocity of first glider if the collision is elastic. Find the direction of the final velocity of first glider if the collision is elastic.
Find the magnitude of the final velocity of second glider if the collision is elastic. Find the direction of the final velocity of second glider if the collision is elastic. Find the final kinetic energy of first glider. Find the final kinetic energy of second glider.
M1 = 0.400kg, V1 = 0.800 m/s.
M2 = 0.150kg, V2 = 0.
M1*V1 + M2*V2 = M1*V3 + M2*V4.
0.4*0.8 + 0.15*0 = 0.4*V3 * 0.15*V4,
Eq1: 0.4*V3 + 0.15*V4 = 0.32.
Conservation of KE Eq:
V3 = (V1(M1-M2) + 2M2*V2)/(M1+M2).
V3 = (0.8(0.4-0.15) + 0.30*0)/(0.4+0.15) = 0.20 /0.55 = 0.364 m/s,rt. = Velocity of M1 after the collision.
In Eq1, replace V3 with 0.364 and solve for V4:
0.4*0.364 + 0.15*V4 = 0.32.
V4 = 1.16 m/s,rt. = Velocity of M2 after the collision.
KE1 = 0.5*M1*V3^2.
KE2 = 0.5*M2*V4^2.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Final velocity of the first glider:
For an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. The momentum is given by mass times velocity.
Initial momentum of the system:
mu1 = m1 * v1 + m2 * v2
= (0.400 kg) * (0.800 m/s) + (0.150 kg) * (0 m/s)
= 0.320 kg·m/s
Since the collision is head-on, the direction of velocity for the second glider is opposite to that of the first glider. Therefore, the equation becomes:
Final momentum of the system:
mf1 = m1 * uf1 + m2 * uf2
Since the second glider is stationary, its final velocity (uf2) can be taken as zero:
mf1 = m1 * uf1 + m2 * 0
uf1 = (mf1 - m2 * 0) / m1
The mass of the first glider is 0.400 kg, and the mass of the second glider is 0.150 kg. The final momentum of the system is the initial momentum since there are no external forces acting on the system:
mf1 = 0.320 kg·m/s
Substituting these values, we get:
uf1 = (0.320 kg·m/s - 0.150 kg * 0) / 0.400 kg
uf1 = 0.800 m/s
Therefore, the magnitude of the final velocity of the first glider is 0.800 m/s.
2. Direction of the final velocity of the first glider:
Since the collision is head-on, the direction of the final velocity for the first glider is in the same direction as its initial velocity, which is to the right.
3. Final velocity of the second glider:
The second glider, being stationary, gains all the momentum initially possessed by the first glider. Therefore, the final velocity of the second glider is equal to the initial velocity of the first glider (0.800 m/s), but in the opposite direction (to the left).
4. Direction of the final velocity of the second glider:
The final velocity of the second glider is to the left, opposite to the initial velocity of the first glider.
5. Final kinetic energy of the first glider:
The kinetic energy (KE) is given by the equation KE = 0.5 * mass * velocity^2.
Initial kinetic energy of the first glider:
KEi1 = 0.5 * m1 * v1^2
= 0.5 * (0.400 kg) * (0.800 m/s)^2
= 0.160 J
Since the collision is elastic, the total kinetic energy of the system remains constant.
Final kinetic energy of the first glider:
KEf1 = 0.160 J
Therefore, the final kinetic energy of the first glider is 0.160 J.
6. Final kinetic energy of the second glider:
The second glider was initially at rest, so its initial kinetic energy is zero. Since the total kinetic energy is conserved, the final kinetic energy of the second glider is also zero.
Therefore, the final kinetic energy of the second glider is 0 J.
To solve this problem, we can use the principles of conservation of momentum and kinetic energy. Let's break it down step by step.
1. Find the magnitude of the final velocity of the first glider if the collision is elastic:
In an elastic collision, both momentum and kinetic energy are conserved. The initial momentum of the system is the sum of the masses of the two gliders multiplied by their respective velocities. The final momentum will also be the sum of the masses multiplied by their final velocities.
Initial momentum = (mass of first glider × velocity of first glider) + (mass of second glider × velocity of second glider)
Since the second glider is stationary, its velocity is 0.
Initial momentum = (0.400 kg × 0.800 m/s) + (0.150 kg × 0) = 0.320 kg·m/s
Since momentum is conserved in an elastic collision, the final momentum will also be 0.320 kg·m/s.
Final momentum = (mass of first glider × final velocity of first glider) + (mass of second glider × final velocity of second glider)
Since we want to find the magnitude of the final velocity for the first glider, we can assign a variable to the final velocity of the second glider.
Final momentum = (0.400 kg × final velocity of first glider) + (0.150 kg × final velocity of second glider)
Since the final momentum is 0.320 kg·m/s, we have the equation:
0.320 kg·m/s = (0.400 kg × final velocity of first glider) + (0.150 kg × final velocity of second glider)
We will use this equation along with the fact that kinetic energy is conserved to solve for the final velocities.
2. Find the direction of the final velocity of the first glider if the collision is elastic:
The direction can be determined based on whether the glider is moving to the left (negative velocity) or right (positive velocity) after the collision.
3. Find the magnitude of the final velocity of the second glider if the collision is elastic:
Since we assigned a variable to the final velocity of the second glider, this will be determined along with the first glider's final velocity when we solve the equation mentioned above.
4. Find the direction of the final velocity of the second glider if the collision is elastic:
Similar to the first glider, the direction can be determined based on whether the glider is moving to the left or right after the collision.
5. Find the final kinetic energy of the first glider:
To find the final kinetic energy of the first glider, we can use the equation:
Final kinetic energy of the first glider = (1/2) × mass of first glider × (final velocity of first glider)^2
Plug in the final velocity of the first glider obtained from the equation in step 1 to find its final kinetic energy.
6. Find the final kinetic energy of the second glider:
Like the first glider, we can use the equation:
Final kinetic energy of the second glider = (1/2) × mass of second glider × (final velocity of second glider)^2
Plug in the final velocity of the second glider obtained from the equation in step 1 to find its final kinetic energy.