A bullet is fired from a rifle, emerging at 340 m/s. It strikes a sandbag some distance away, having lost 10% of its velocity due to air resistance. if it penetrates the sandbag to a depth of 12 meters, how long did it take for the bullet to come to rest in the bag?
To find the time it took for the bullet to come to rest in the sandbag, we can use the equation of motion that relates velocity and distance traveled.
First, we need to determine the initial velocity of the bullet before it lost 10% of its velocity. Since the bullet emerged from the rifle at a velocity of 340 m/s, we can calculate the initial velocity as:
Initial velocity = 340 m/s
Since the bullet lost 10% of its velocity due to air resistance, the final velocity can be calculated as:
Final velocity = Initial velocity - (10% of Initial velocity)
Final velocity = 340 m/s - (0.1 * 340 m/s)
Final velocity = 340 m/s - 34 m/s
Final velocity = 306 m/s
Next, we can use the equation of motion to find the time it took for the bullet to come to rest. The equation is as follows:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance
Since the bullet comes to rest, the final velocity is 0 m/s. We can substitute the known values in the equation and solve for time:
0^2 = 306^2 + 2 * acceleration * 12
Simplifying this equation gives us:
0 = 306^2 + 24 * acceleration
We need to solve for acceleration. Rearranging the equation, we have:
acceleration = -(306^2 / 24)
Plugging in the values and calculating acceleration, we get:
acceleration = -(306^2 / 24)
acceleration ≈ -3970.5 m/s^2
Since the acceleration is negative, it indicates deceleration. Now, we can find the time taken by using the equation:
Final velocity = Initial velocity + (acceleration * time)
Substituting values, we have:
0 m/s = 306 m/s + (-3970.5 m/s^2) * time
Simplifying this equation, we get:
-306 m/s = -3970.5 m/s^2 * time
Solving for time gives us:
time = -306 m/s / (-3970.5 m/s^2)
time ≈ 0.077 s
Therefore, it took approximately 0.077 seconds for the bullet to come to rest in the sandbag.