If a body loses half of its velocity on penetrating a 4cm wooden block how much more will it penetrate before coming to rest
To solve this question, we need to understand the principle of conservation of mechanical energy.
According to the principle of conservation of energy, the initial kinetic energy of the body will be equal to the sum of the final kinetic energy and the work done against friction during the penetration.
Let's assume the initial velocity of the body is v₀, and it loses half of its velocity while penetrating the 4cm wooden block. Therefore, the final velocity (v) will be v₀/2.
The work done against friction can be calculated using the formula:
Work = Force × distance
The force required to stop the body is equal to the force of friction. This force can be calculated using Newton's second law:
Force = mass × acceleration
Considering the acceleration due to the body coming to rest, the force of friction is given by:
Force = mass × deceleration
Based on Newton's second law, we also know that deceleration is given by:
deceleration = (final velocity - initial velocity) / time
As the body comes to rest, the final velocity will be zero, so:
deceleration = -v₀ / (t/2)
Since the body goes through 4cm of wood, we can assume that the time it takes is proportional to the length traveled. Therefore:
t/2 = 4cm / v₀
Wrapping up, using the above information, we can set up the equation to solve for v₀:
1/2 * (1/2 * m * v₀²) = m * (-v₀ / (4cm/v₀)) * 4cm
Simplifying and rearranging the equation, we get:
1/8 * v₀² = 4cm²/v₀
Cross-multiplying, we have:
v₀³ = 32cm³
Taking the cube root of both sides, we find:
v₀ = 3.1748 cm/s
Now, we can calculate the additional distance the body will penetrate before coming to rest. Let's assume the additional distance is x cm.
Using the equation for deceleration, we have:
deceleration = (final velocity - initial velocity) / time
Since the final velocity is 0 and the initial velocity is v₀, we have:
deceleration = -v₀ / t
Now, we can use the equation of motion:
x = initial velocity * time + (1/2) * acceleration * time²
Substituting the values, we get:
x = (v₀) * (t) + (1/2) * (-v₀/t) * t²
Simplifying further:
x = v₀ * t - (1/2) * v₀ * t
x = (1/2) * v₀ * t
Substituting the known values:
x = (1/2) * 3.1748 cm/s * (4cm / (3.1748 cm/s))
x ≈ 2 cm
Therefore, the body will penetrate approximately 2 cm more before coming to rest.