A rifle is aimed horizontally at a target 31 m away. The bullet hits the target 1.3 cm below the aiming point. Neglect air resistance.
(a) What is the bullet's time of flight?
s
(b) What is its muzzle velocity?
m/s (in the bullet's initial horizontal direction)
a. d = 0.5g*t^2 = 0.013 m.
4.9t^2 = 0.013t^2
t^2 = 0.00265
Tf = 0.052 s. = Fall time or time in flight.
b. Xo * T = 31 m.
0.052Xo = 31
Xo = 602 m/s.
x=x0+v0xt+(1/2)axt^2
=> -0.013~0+0+(1/2)(-9.8)t^2
=> 0.013~(1/2)(9.8)t^2
=> 0.013~4.9t^2
=> t~0.052s
y=y0+v0yt+(1/2)ayt^2
=> 31~0+v0yt+(1/2)(0)(0.052)^2
=> 31~v0yt
=>v0y~31/t
=>v0y~31/.052
=>v0y~6.0*10^2m/s
To find the bullet's time of flight, we can use the fact that the horizontal component of its velocity remains constant throughout its motion. Neglecting air resistance, this means that the bullet's horizontal displacement is equal to its initial horizontal velocity multiplied by the time of flight.
(a) The horizontal displacement is given by the distance between the rifle and the target, which is 31 m. We can convert this distance to centimeters for consistency: 31 m = 3100 cm.
Since the bullet hits 1.3 cm below the aiming point, we can subtract this value from the total distance. Therefore, the horizontal displacement is actually 3100 cm - 1.3 cm = 3098.7 cm.
Now we need to find the bullet's initial horizontal velocity. Since the rifle is aimed horizontally, there is no initial vertical velocity. Therefore, the bullet's initial velocity is purely horizontal.
To find the time of flight, we can use the equation:
horizontal displacement = initial horizontal velocity × time of flight
It is important to note that the bullet travels in a parabolic trajectory, so solving for time using this equation provides the time at the midpoint of the bullet's journey. This is because the time it takes for the bullet to reach the target is equal to the time it takes to reach the target when it is at the highest point of its trajectory.
Using the equation, we have:
3098.7 cm = initial horizontal velocity × time of flight
Since the bullet's trajectory is symmetric, the time of flight can be found by dividing the total time it takes for the bullet to reach the target by 2.
Therefore, the bullet's time of flight is:
time of flight = (3098.7 cm / initial horizontal velocity) / 2
Now, let's move on to finding the muzzle velocity.
(b) The bullet's muzzle velocity refers to its initial velocity when it is fired from the rifle. Since we are neglecting air resistance, the vertical component of the bullet's velocity remains constant throughout its motion. This means that the bullet's initial vertical velocity is zero.
The initial velocity of the bullet can be decomposed into its horizontal and vertical components. The horizontal component remains constant at the muzzle velocity, while the vertical component is 0.
Using the Pythagorean theorem, we can find the muzzle velocity:
muzzle velocity = sqrt((horizontal component of initial velocity)^2 + (vertical component of initial velocity)^2)
Since the vertical component of the initial velocity is 0, the muzzle velocity simplifies to just the horizontal component:
muzzle velocity = horizontal component of initial velocity
Therefore, the muzzle velocity is equal to the initial horizontal velocity we found earlier.
To summarize:
(a) The bullet's time of flight is given by the equation time of flight = (3098.7 cm / initial horizontal velocity) / 2.
(b) The muzzle velocity is equal to the initial horizontal velocity.