A ball is attached to one
end of a wire, the other end being
fastened to the ceiling. The wire
is held horizontal, and the ball is
released from rest (see the drawing).
It swings downward and
strikes a block initially at rest on
a horizontal frictionless surface.
Air resistance is negligible, and
the collision is elastic. The masses of the ball and block are, respectively,
1.60 kg and 2.40 kg, and the length of the wire is 1.20 m. Find the velocity
(magnitude and direction) of the ball (a) just before the collision, and
(b) just after the collision.
6400 N
To find the velocity of the ball just before the collision, we can use the principle of conservation of mechanical energy. Since the ball is released from rest, its initial potential energy due to its height is converted into kinetic energy just before the collision.
Step 1: Calculate the potential energy of the ball just before the collision.
Potential energy = mass * acceleration due to gravity * height
Potential energy = 1.60 kg * 9.8 m/s^2 * 1.20 m
Potential energy = 18.816 J
Step 2: Convert the potential energy to kinetic energy.
Since the ball is released from rest, the initial kinetic energy is zero. So, the total mechanical energy just before the collision is equal to the potential energy calculated in step 1.
Step 3: Find the velocity just before the collision.
Total mechanical energy = kinetic energy
Kinetic energy = 18.816 J
Since the formula for kinetic energy is given by:
Kinetic energy = (1/2) * mass * velocity^2
Rearrange the formula to solve for velocity:
velocity = square root((2 * kinetic energy) / mass)
velocity = square root((2 * 18.816 J) / 1.60 kg)
velocity ≈ 6.97 m/s
Therefore, the velocity of the ball just before the collision is approximately 6.97 m/s in the downward direction.
To find the velocity of the ball just after the collision, we can apply the principle of conservation of momentum.
Step 1: Calculate the initial momentum of the system (ball and block) just before the collision.
Initial momentum = mass of the ball * velocity of the ball + mass of the block * velocity of the block
Since the block is initially at rest, its velocity is zero, thus simplifying the equation:
Initial momentum = mass of the ball * velocity of the ball
Initial momentum = 1.60 kg * 6.97 m/s
Step 2: Calculate the final momentum of the system just after the collision.
Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision.
Step 3: Find the velocity of the ball after the collision.
Final momentum = mass of the ball * velocity of the ball after the collision
Rearrange the equation to solve for velocity:
velocity after collision = final momentum / mass of the ball
velocity after collision = initial momentum / mass of the ball
velocity after collision = (1.60 kg * 6.97 m/s) / 1.60 kg
velocity after collision ≈ 6.97 m/s
Therefore, the velocity of the ball just after the collision is approximately 6.97 m/s in the downward direction.