You demonstrate hitting a board in a karate class. The speed of your hand as it hits the thick board is 20 m/s with respect to the board, and the mass of your hand is about 0.82 kg. How deep does your hand go into the board before stopping if the collision lasts for 2.0×10−3s?
V = Vo + a*t = 0.
20 + a*0.002 = 0,
a = -10,000 m/s^2.
V^2 = Vo^2 + za*d = 0.
20^2 + (-20,000)d = 0,
d = 0.02 m.
To determine how deep your hand goes into the board, we can use the principles of conservation of momentum and energy.
Here's how you can calculate the depth:
Step 1: Find the impulse applied to the board.
Impulse (J) = Change in momentum (Δp)
Since momentum (p) is mass (m) multiplied by velocity (v), we can calculate the initial momentum (p_initial) and the final momentum (p_final) of your hand.
p_initial = m * v_initial
p_final = m * v_final
The initial momentum is given as:
p_initial = 0.82 kg * 20 m/s
Since the hand comes to rest after hitting the board, the final momentum is zero:
p_final = 0
So, the change in momentum (Δp) is:
Δp = p_final - p_initial
= 0 - (0.82 kg * 20 m/s)
Step 2: Calculate the impulse.
The impulse applied to the board is equal to the change in momentum:
J = Δp
Step 3: Use the equation for impulse.
Impulse (J) = Force (F) * Time (Δt)
Rearranging the equation, we can solve for force (F):
F = J / Δt
Plug in the values:
F = Δp / Δt
Step 4: Calculate the force exerted on the board.
The force exerted on the board is equal to the impulse divided by the collision time:
F = Δp / Δt
Step 5: Find the depth of penetration.
The depth of penetration is related to the average force applied to the board during the collision and the time of collision.
The average force (F_avg) can be calculated using:
F_avg = F = Δp / Δt
The depth (d) is related to the average force, density (ρ) of the board, and the area of contact (A) between your hand and the board:
d = (F_avg * Δt^2) / (2 * ρ * A)
Note: It is important to note that the area of contact between the hand and the board is crucial and can significantly affect the depth of penetration.
By plugging in the known values, such as the impulse, area of contact, and density, you can calculate the depth to which your hand goes into the board.