a bullet fired into a target loses half its velocity after penetrating 25cm.How much further will it penetrate before coming to rest ?
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To solve this question, we can use the concept of constant deceleration due to an external force (usually air resistance) acting on the bullet.
Let's assume that the initial velocity of the bullet is "v" and it loses half of its velocity after penetrating 25 cm.
We can calculate the deceleration "a" using the formula:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the bullet comes to rest)
u = initial velocity (v)
a = deceleration
s = displacement (25 cm or 0.25 m)
Rearranging the formula:
a = (v^2 - u^2) / (2s)
Substituting the given values:
a = (0 - v^2) / (2 * 0.25)
Simplifying:
a = -v^2 / 0.5
a = -2v^2
Since the acceleration is negative, it indicates deceleration.
Now, we will use the equation of motion:
v^2 = u^2 + 2as
Substituting the given values:
0 = v^2 + 2 * (-2v^2) * s
0 = v^2 - 4v^2s
Rearranging the equation:
v^2(1 - 4s) = 0
Since velocity cannot be zero, we can conclude that 1 - 4s = 0.
So, 1 = 4s
Therefore, the bullet will penetrate an additional distance of:
s' = (1 / 4) * s
s' = (1 / 4) * 0.25
s' = 0.0625 meters or 6.25 centimeters
Hence, the bullet will penetrate approximately an additional 6.25 centimeters before coming to rest.
To find out how much further the bullet will penetrate before coming to rest, we first need to understand the concept of deceleration.
When a bullet penetrates a target, it experiences a deceleration or a decrease in velocity due to the opposing forces acting upon it, such as air resistance and internal friction. In this case, the bullet loses half its velocity after penetrating a distance of 25 cm.
To determine the additional distance the bullet will travel before coming to rest, we can use the formula of motion with uniform deceleration:
V² = U² - 2as
Where:
V = Final velocity (which will be zero when the bullet comes to rest)
U = Initial velocity (half the original velocity of the bullet)
a = Acceleration (which is the deceleration in this case, since the bullet is slowing down)
s = Distance traveled (what we want to find)
Let's break it down step by step:
Step 1: Find the values
Given that the initial velocity of the bullet is halved after penetrating 25 cm, we can assume the initial velocity (U) is equal to 0.5 times the original velocity.
The final velocity (V) is zero since it comes to rest.
The acceleration (a) is negative because it acts in the opposite direction of the bullet's initial motion.
So, V = 0 m/s
U = 0.5v (where v is the original velocity)
a = -?
Step 2: Calculate the acceleration
We need to find the exact value of the deceleration (a) acting on the bullet. To do this, we can use the concept of conservation of mechanical energy. When the bullet penetrates the target and comes to rest, all of its kinetic energy is converted into other forms of energy (such as heat and sound). Therefore, we can equate the initial kinetic energy (K) to the work done by the decelerating force(s).
K = Work
(1/2) mu² = Fd
m = mass of the bullet
u = initial velocity (0.5v)
F = force decelerating the bullet
d = distance traveled (25 cm converted into meters = 0.25 m)
Step 3: Substitute the values and solve for acceleration
Substituting the known values into the equation, we can solve for the force (F):
(1/2) mu² = Fd
(1/2) m(0.5v)² = F(0.25)
(1/8)mv² = F(0.25)
Since the force (F) is equal to the product of the mass (m) and the acceleration (a), we can rewrite the equation as:
(1/8)ma² = ma(0.25)
(1/8)a² = (1/4)
a² = 2
Taking the square root of both sides, we find that:
a = √2 m/s²
Now that we know the acceleration, we can proceed to the next step.
Step 4: Calculate the additional distance
Using the formula of motion with uniform deceleration (V² = U² - 2as), we can solve for the additional distance traveled (s):
0² = (0.5v)² - 2(√2)s
0 = 0.25v² - 2(√2)s
2(√2)s = 0.25v²
s = (0.25/2√2)v²
s = (v²/8√2)
Therefore, the bullet will penetrate an additional distance of (v²/8√2) units before coming to rest.
Remember, in this calculation, v represents the original velocity of the bullet.
Kinetic energy, KE = (1/2)mv²
resistive force = constant
Retaining half the velocity means (1/2)^2=1/4 of the kinetic energy.
Since 50% of the KE penetrates 25 cm, so
25% of the (remaining) KE penetrates 12.5 cm.