A rifle is aimed horizontally at a target 50 m away. The bullet hits the target 2.4 cm below the aim point.
Part A
What was the bullet's flight time? Neglect the air resistance
Part B
What was the bullet's speed as it left the barrel?
In the absence of air resistance we can split the movement of the bullet into the vertical and horizontal components.
The former is a free-fall.
Using the kinematics equation below, we can solve for the time it takes to free-fall through 2.4 cm:
S=(v0)t+(1/2)at^2
with
v0=initial (vertical) velocity = 0
a=acceleration due to gravity=-9.81 m/s^2
s=-0.024 m
solve for t to get
t=sqrt(2*0.024/9.81)=0.07 s
If the bullet covered 50m in 0.07s, you can calculate the bullet's muzzle speed, in m/s.
To find the bullet's flight time, we can use the formula for horizontal motion:
d = v * t
where:
d is the horizontal distance (50 m)
v is the horizontal velocity
t is the flight time (what we want to find)
Since the bullet is aimed horizontally, the horizontal velocity is the same throughout the flight.
For Part A, let's find the flight time of the bullet:
Step 1: Convert the horizontal distance to centimeters:
50 m = 5000 cm
Step 2: Substitute the values into the formula and rearrange it to solve for t:
5000 cm = v * t
Step 3: Rearrange the formula to solve for t:
t = 5000 cm / v
Now let's move on to Part B and find the bullet's speed as it left the barrel:
Since air resistance is neglected, we can assume that the only force acting on the bullet is gravity, causing it to fall 2.4 cm below the aim point.
Using the formula for vertical motion under constant acceleration:
d = (1/2) * g * t^2
where:
d is the vertical distance (2.4 cm)
g is the acceleration due to gravity (9.8 m/s^2, assuming the units for d were also in meters)
t is the flight time (what we found in Part A)
Step 4: Convert the vertical distance to meters:
2.4 cm = 0.024 m
Step 5: Substitute the values into the formula:
0.024 m = (1/2) * 9.8 m/s^2 * t^2
Step 6: Simplify the equation:
0.024 m = 4.9 m/s^2 * t^2
Step 7: Solve for t by taking the square root of both sides:
t = sqrt(0.024 m / 4.9 m/s^2)
Now that we have found the value for t, we can substitute it back into the equation from Part A to find the bullet's speed as it left the barrel:
t = 5000 cm / v
Step 8: Convert the value of t to seconds (to match the units for velocity):
t = sqrt(0.024 m / 4.9 m/s^2) * (1 m / 100 cm) = sqrt(0.024 m / 4.9 m/s^2) * 0.01 s/cm
Step 9: Substitute the value of t back into the equation:
sqrt(0.024 m / 4.9 m/s^2) * 0.01 s/cm = 5000 cm / v
Step 10: Rearrange the formula to solve for v:
v = 5000 cm / (sqrt(0.024 m / 4.9 m/s^2) * 0.01 s/cm)
Simplifying the equation gives us the bullet's speed as it left the barrel.
Please note that this calculation assumes the bullet's horizontal velocity remains constant throughout its flight and that air resistance is neglected.
To determine the bullet's flight time and speed, we can use the equations of motion. Let's break down the problem step by step:
Part A: Finding the flight time
To find the flight time, we can use the horizontal motion of the bullet. Since the rifle is aimed horizontally and there is no air resistance, the horizontal velocity remains constant throughout the motion.
The horizontal distance traveled by the bullet is given as 50 m. We can use the formula:
Horizontal distance (d) = Velocity (v) * Time (t)
Rearranging the formula, we get:
Time (t) = Horizontal distance (d) / Velocity (v)
Given the horizontal distance d = 50 m, we need to determine the velocity v.
Part B: Finding the initial velocity (speed)
To find the bullet's speed as it left the barrel, we can use the vertical motion of the bullet. We know that the bullet hits the target 2.4 cm (or 0.024 m) below the aim point.
Using the vertical motion equations, we can use the formula for the vertical displacement:
Vertical displacement (y) = Initial vertical velocity (u) * Time (t) + (0.5 * Acceleration (a) * Time (t)^2)
Since the bullet is fired horizontally, the initial vertical velocity (u) is zero, and we can ignore the second term in the equation. This simplifies the equation to:
Vertical displacement (y) = 0.5 * Acceleration (a) * Time (t)^2
Given the vertical displacement y = 0.024 m, we need to determine the time t.
To find the acceleration, we can use the fact that the only force acting on the bullet vertically is gravity. The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Now, let's solve for time (t) in both equations:
For Part A:
Time (t) = Horizontal distance (d) / Velocity (v)
For Part B:
Vertical displacement (y) = 0.5 * Acceleration (a) * Time (t)^2
By solving both equations simultaneously, we can determine the flight time (t) and the velocity (v).