You demonstrate hitting a board in a karate class. The speed of your hand as it hits the thick board is 14 m/s with respect to the board, and the mass of your hand is about .80 kg. How deep does your hand go into the board before stopping if the collision lasts for 2. X 10^-3 s?
average speed during the stopping:
avg speed=14/2 m/s
depth=speed*time=7E-3 m or .007 meters.
To find out how deep your hand goes into the board, we can use the principles of Newton's laws of motion and calculate the impulse experienced by your hand during the collision.
First, we need to calculate the impulse. Impulse is defined as the change in momentum of an object during a collision. It can be calculated using the formula:
Impulse = Force × Time
However, since we are given the speed of your hand with respect to the board and its mass, we need to use the equation:
Impulse = Change in Momentum = Mass × (Final Velocity - Initial Velocity)
Here, the final velocity of your hand is 0 m/s because it stops after the collision. The initial velocity is given as 14 m/s, and the mass of your hand is 0.80 kg.
Impulse = 0.80 kg × (0 m/s - 14 m/s)
Impulse = 0.80 kg × (-14 m/s)
Impulse = -11.2 kg⋅m/s (Note: The negative sign indicates the direction of the force exerted by the board on your hand.)
Now that we have the impulse, we can determine how deep your hand goes into the board. Let's assume the depth is represented by 'd'. The impulse can also be defined as the force exerted during the collision multiplied by the time of collision:
Impulse = Force × Time
-11.2 kg⋅m/s = Force × (2.0 × 10^-3 s)
Solving for the force:
Force = -11.2 kg⋅m/s ÷ (2.0 × 10^-3 s)
Force ≈ -5,600 N
Considering that the force exerted by the board is directed opposite to the motion of your hand, we take the magnitude of the force without the negative sign.
Finally, we can use Hooke's Law to determine the depth your hand goes into the board. Hooke's Law states that the force exerted on an object is directly proportional to the depth the object goes into the material, assuming the material obeys Hooke's Law.
Hooke's Law: Force = k × Depth
Where 'k' is the proportionality constant, which depends on the material properties.
Since we have the force exerted on your hand (approximately 5,600 N), we can rearrange Hooke's Law to solve for the depth:
Depth = Force ÷ k
The value of 'k' depends on the specific material used for the board. Without that information, we cannot provide an exact value for the depth.