Question

In a first experiment, a 40-g clay ball is shot at a speed of 1.3 m/s horizontally from the edge of a table. The ball lands on the floor 0.60 m from the table. In a second experiment, the same ball is shot at the same speed, but this time the ball hits a wooden block that is placed on the edge of the table. The ball sticks to the block, and the block lands on the floor 70 mm from the table.

A> Determine the height of the table.
B> Determine the mass of the block.

Answer 1

Answer: 0.94 kg.

Explanation: A> To solve for the height of the table, we can use the kinematic equation for horizontal motion:

distance = velocity × time

In the first experiment, the ball travels a distance of 0.60 m horizontally before hitting the ground. We know the initial horizontal velocity of the ball is 1.3 m/s, and we can assume there is negligible air resistance, so the time of flight can be found using:

time = distance / velocity

time = 0.60 m / 1.3 m/s

time ≈ 0.46 s

We can use the kinematic equation for vertical motion to find the height of the table:

height = initial vertical velocity × time + 0.5 × acceleration × time^2

Since the ball starts from rest vertically and lands on the ground, the final vertical velocity is also zero. We can assume the acceleration due to gravity is -9.81 m/s^2 (negative because it acts in the opposite direction to the initial velocity). Plugging in the known values:

height = 0 + 0.5 × (-9.81 m/s^2) × (0.46 s)^2

height ≈ 0.49 m

Therefore, the height of the table is approximately 0.49 m.

B> To solve for the mass of the block, we can use conservation of momentum. Before the collision, only the clay ball is moving horizontally with a velocity of 1.3 m/s. After the collision, the clay ball and the block move together with a common velocity. Since the horizontal direction is the same before and after the collision, we can write:

momentum before = momentum after

(mass of clay ball) × (initial velocity of clay ball) = (mass of clay ball + mass of block) × (common velocity after collision)

Plugging in the known values:

(0.04 kg) × (1.3 m/s) = (0.04 kg + mass of block) × (velocity after collision)

We also know from the second experiment that the block and ball together travel a distance of 0.07 m (70 mm) horizontally before hitting the ground. We can use this to find the common velocity after the collision using:

distance = velocity × time

time = distance / velocity

time = 0.07 m / (1.3 m/s)

time ≈ 0.054 s

velocity after collision = distance / time

velocity after collision = 0.07 m / 0.054 s

velocity after collision ≈ 1.3 m/s

Plugging in this value and solving for the mass of the block:

(0.04 kg) × (1.3 m/s) = (0.04 kg + mass of block) × (1.3 m/s)

mass of block ≈ 0.94 kg

Therefore, the mass of the block is approximately 0.94 kg.

Answer 2

The height of the table is approximately 1.06 meters. The mass of the block is approximately 0.17 kg.

To determine the height of the table in the first experiment, we can use the equation for horizontal motion: distance = velocity x time

Since the clay ball was shot horizontally and landed on the floor 0.60 m from the table, the distance is 0.60 m. The velocity is 1.3 m/s. We can rearrange the equation to solve for time:

time = distance / velocity

Substituting the values, we get:

time = 0.60 m / 1.3 m/s = 0.46 s

The height of the table can be found using the equation for vertical motion:

height = 0.5 x acceleration x time^2

Since the acceleration is due to gravity and is approximately 9.8 m/s^2, we can substitute the values to calculate the height:

height = 0.5 x 9.8 m/s^2 x (0.46 s)^2 = 1.06 m

Therefore, the height of the table is approximately 1.06 meters.

To determine the mass of the block in the second experiment, we can use the principle of conservation of momentum:

momentum before = momentum after

Since the clay ball sticks to the block, the momentum before impact is given by:

momentum before = mass of the clay ball x velocity of the clay ball

Since the velocity of the clay ball is 1.3 m/s, we can substitute the values:momentum before = 0.04 kg x 1.3 m/s = 0.052 kg m/s

The momentum after the impact is given by:

momentum after = (mass of the clay ball + mass of the block) x velocity of the block

Since the velocity of the block is unknown, we need to find it using the distance the block travels. The distance is given as 70 mm or 0.07 m. The time it takes for the block to reach the floor is the same as the time it took for the clay ball to reach the floor, which is 0.46 s. We can use the equation for horizontal motion to find the velocity:

velocity = distance / time

Substituting the values, we get:

velocity = 0.07 m / 0.46 s = 0.15 m/s

Now we can substitute the values into the equation for momentum after:

momentum after = (0.04 kg + mass of the block) x 0.15 m/s

Since the momentum before and after the impact are equal, we can set up the equation:0.052 kg m/s = (0.04 kg + mass of the block) x 0.15 m/s

Solving for the mass of the block:

mass of the block = (0.052 kg m/s) / (0.15 m/s) - 0.04 kg = 0.17 kg

Therefore, the mass of the block is approximately 0.17 kg.