19. A track consists of a frictionless incline plane, which is a height of 0.5 m, and a rough horizontal section with a coefficient of kinetic friction 0.02. Block A, whose mass is1.5 kg, is released from the top of the incline plane, slides down and collides instantaneously and in elastically with identical block B at the lowest point. The two blocks move to the right through the rough section of the track until they stop.
a. Determine the initial potential energy of block A.
b. Determine the kinetic energy of block A at the lowest point, just before the collision.
c. Find the speed of the two blocks just after the collision.
d. Find the kinetic energy of the two blocks just after the collision.
e. How far will the two blocks travel on the rough section of the track?
f. How much work will the friction force do during this time?
M*g = 1.5 * 9.8 = 14.7 N.=Wt. of blk A.
a. PE = M*g*h = 14.7*0.5 = 7.35 J.
b. V^2 = Vo^2 + 2g*h = 0 + 19.6*0.5 = 9.8.
3.13 m/s.
KE = 0.5M*V^2 = 0.5*1.5*3.13^2 = 7.35 J.
c. M1*V1 + M2*V2 = M1*V + M2*V.
1.5*3.13 + 1.5*0 = 1.5V + 1.5V.
4.7 + 0 = 3V.
V = 1.57 m/s.
d. KEa = KEb = 0.5*1.5*1.57^2 = 1.85 J.
2
a. The initial potential energy of block A can be calculated using the formula:
Potential energy = mass * gravity * height
Given:
mass of block A (m) = 1.5 kg
height (h) = 0.5 m
gravity (g) = 9.8 m/s^2 (acceleration due to gravity)
Plugging in the values:
Potential energy = 1.5 kg * 9.8 m/s^2 * 0.5 m
Potential energy = 7.35 J
So, the initial potential energy of block A is 7.35 Joules.
b. The kinetic energy of block A at the lowest point, just before the collision can be calculated using the formula:
Kinetic energy = 0.5 * mass * velocity^2
Since block A is sliding down the incline plane with no mention of other forces acting on it, we can assume it's only the gravitational potential energy that is converted to kinetic energy.
So, the initial potential energy of block A (7.35 Joules) is equal to its initial kinetic energy.
c. To determine the speed of the two blocks just after the collision, we need to apply the principle of conservation of linear momentum.
Since block B is identical to block A, they have the same mass (1.5 kg) and collide inelastically, the total momentum before the collision is the same as the total momentum after the collision.
Momentum before collision = Momentum after collision
The momentum of an object is given by the equation:
Momentum = mass * velocity
Let the initial velocity of block A be Va and the initial velocity of block B be Vb.
So, the initial momentum before the collision is:
Momentum before collision = mass of A * velocity of A + mass of B * velocity of B
= 1.5 kg * Va + 1.5 kg * Vb
The momentum after the collision is:
Momentum after collision = (mass of A + mass of B) * velocity after collision
= 1.5 kg * Va' + 1.5 kg * Vb'
Since the two blocks stick together and move with the same velocity after the collision, we can assume Va' = Vb'.
So, the momentum after the collision is:
Momentum after collision = 3 kg * Va'
Setting the momentum before and after the collision equal to each other, we have:
1.5 kg * Va + 1.5 kg * Vb = 3 kg * Va'
Va + Vb = 2 Va'
Now, we need to consider the coefficient of restitution (e) for an inelastic collision, which is given as:
e = (velocity of separation)/(velocity of approach)
For an inelastic collision, e is less than 1, and the two blocks stick together after the collision. Therefore, the velocity of separation is zero.
So, the equation becomes:
e = 0 / (Va' - Va)
Va' - Va = 0
Va' = Va
Therefore, the speed of the two blocks just after the collision is the same as the speed of block A just before the collision, which is Va.
d. The kinetic energy of the two blocks just after the collision can be calculated using the formula:
Kinetic energy = 0.5 * (mass of A + mass of B) * velocity^2
Given:
mass of A (mA) = mass of B (mB) = 1.5 kg
velocity of the two blocks after collision (V') = Va
Plugging in the values:
Kinetic energy = 0.5 * (1.5 kg + 1.5 kg) * Va^2
Kinetic energy = 0.5 * 3 kg * Va^2
Kinetic energy = 1.5 kg * Va^2
e. The distance the two blocks travel on the rough section of the track can be determined using the work-energy principle.
The work done by the friction force is given by the equation:
Work = force of friction * distance
The force of friction can be calculated using the formula:
Force of friction = coefficient of kinetic friction * normal force
On the horizontal section of the track, the normal force is equal to the weight of the two blocks, since there is no vertical acceleration.
Normal force = (mass of A + mass of B) * gravity
Plugging in the values:
Normal force = 3 kg * 9.8 m/s^2
Normal force = 29.4 N
The force of friction is:
Force of friction = coefficient of kinetic friction * normal force
= 0.02 * 29.4 N
= 0.588 N
The work done by the friction force is equal to the change in kinetic energy of the two blocks.
Initial kinetic energy - Final kinetic energy = Work
Since the blocks come to a stop, the final kinetic energy is zero.
So, we have:
1.5 kg * Va^2 - 0 = 0.588 N * distance
Distance = (1.5 kg * Va^2) / (0.588 N)
Plugging in the given value of Va, we can calculate the distance traveled by the two blocks.
f. The work done by the friction force is given by the equation:
Work = force of friction * distance
Using the force of friction calculated in part (e) and the distance traveled, we can determine the work done by the friction force.
To answer these questions, we need to apply various equations and principles of physics. Let's go step by step.
a. To determine the initial potential energy of block A, we need to find the height from which it is released. In this case, the height is given as 0.5 m. The potential energy is given by the equation PE = mgh, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height. Plugging in the values, we get PE = 1.5 kg * 9.8 m/s^2 * 0.5 m = 7.35 J.
b. To determine the kinetic energy of block A at the lowest point just before the collision, we need to consider that all the potential energy is converted into kinetic energy. So the kinetic energy at the lowest point is equal to the initial potential energy. Therefore, the kinetic energy of block A is also 7.35 J.
c. To find the speed of the two blocks just after the collision, we can use the principle of conservation of momentum. In an inelastic collision, the total momentum is conserved. The initial momentum is the mass of block A times its velocity (initially 0 since it was at rest) plus the mass of block B times its velocity (which we need to find). The final momentum is the sum of their masses times their common velocity after the collision. Mathematically, we can write it as:
(mA * 0) + (mB * vB) = (mA + mB) * v
Simplifying, we get v = (mB * vB) / (mA + mB).
Since both blocks are identical and their mass is 1.5 kg, we can substitute:
v = (1.5 kg * vB) / (1.5 kg + 1.5 kg) = vB / 2.
Therefore, the speed of the two blocks just after the collision is half the speed of block B.
d. The kinetic energy of the two blocks just after the collision can be calculated using the equation KE = (1/2) * m * v^2, where m is the mass of the block and v is its velocity. Since both blocks have the same mass, we can calculate the kinetic energy of a single block just after the collision and then double it. The kinetic energy of block B just after the collision is KE = (1/2) * 1.5 kg * (vB/2)^2.
Simplifying, we get KE = 0.375 * vB^2.
Since we have two identical blocks, the total kinetic energy of both blocks is twice this value, so it becomes KE = 2 * 0.375 * vB^2.
e. To determine how far the two blocks will travel on the rough section of the track, we need to consider the work done by the friction force. The work done by friction is equal to the force of friction multiplied by the distance over which it acts. The force of friction can be calculated using the equation F = μ * N, where μ is the coefficient of kinetic friction and N is the normal force (equal to the weight in this case, since the track is horizontal). The weight of a block is given by the equation W = m * g.
The work done by friction is then given by the equation W = F * d, where d is the distance.
Since both blocks have the same mass, weight, and coefficient of kinetic friction, we can calculate the work done by friction for a single block and then double it.
W = 2 * (μ * m * g * d).
We need to find the distance traveled, so we rearrange the equation and solve for d:
d = W / (2 * μ * m * g).
f. The work done by the friction force during the time the blocks are traveling on the rough section of the track is equal to the negative of the change in kinetic energy. In other words, it is the initial kinetic energy minus the final kinetic energy.
The initial kinetic energy is the same as the kinetic energy just after the collision, which we calculated in part (d). The final kinetic energy is zero since the blocks come to a stop.
Therefore, the work done by the friction force is equal to the initial kinetic energy of the two blocks just after the collision.