A rifle is aimed horizontally at a target 30m away. The bullet hits the target 1.9 cm below

the aiming point. (a) What is the bullet’s time of flight? (b) What is the muzzle velocity?

how long does it take (t) for gravity to drop the bullet?

1/2 * 980 cm/s^2 * t^2 = 1.9 cm ... solve for t

(b) the target distance divided by the flight time gives the AVERAGE velocity
... should be close to the muzzle velocity, but not the same

a) y=-1/2 gt^2

so, t=root((-2)*(-0.019m)/9.8m/s^2)=6.2x10^-2s
b)Muzzle velocity =x/t=30m/6.2x10^-2s=4.8x10^-2 m/s

To solve this problem, we can use the equations of motion for uniformly accelerated linear motion. Let's break it down step-by-step:

Step 1: Determine the initial vertical velocity of the bullet.

Since the rifle is aimed horizontally, the initial vertical velocity (Vy) of the bullet is zero. This is because there is no vertical component of the initial velocity.

Step 2: Determine the time of flight.

We can use the equation of motion in the vertical direction to find the time of flight (t). The equation is:

Δy = Vy * t + (1/2) * g * t^2

Where:
Δy is the vertical displacement (the bullet's drop of 1.9 cm in this case)
Vy is the initial vertical velocity (which is zero)
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Plugging in the values and solving for t:

0.019m = 0 * t + (1/2) * 9.8 m/s^2 * t^2

0.019m = 4.9 m/s^2 * t^2

t^2 = 0.019m / (4.9 m/s^2)

t^2 ≈ 0.00387755

t ≈ √(0.00387755) ≈ 0.0623 s

So, the time of flight is approximately 0.0623 seconds.

Step 3: Determine the horizontal velocity of the bullet.

We know that the horizontal displacement (Δx) is equal to 30m. We can use the equation of motion in the horizontal direction to find the horizontal velocity (Vx) of the bullet. The equation is:

Δx = Vx * t

Plugging in the values and solving for Vx:

30m = Vx * 0.0623s

Vx = 30m / 0.0623s

Vx ≈ 481.71 m/s

So, the horizontal velocity, or the muzzle velocity of the bullet, is approximately 481.71 m/s.

To summarize:
(a) The bullet's time of flight is approximately 0.0623 seconds.
(b) The muzzle velocity of the bullet is approximately 481.71 m/s.

To find the bullet's time of flight and muzzle velocity, we can use the equations of motion for projectiles. Let's break it down step by step:

(a) Finding the bullet's time of flight:
The bullet was fired horizontally, so it only experiences vertical motion due to gravity. We can use the equation:

Δy = V₀y * t + (1/2) * g * t²

Since the bullet is aimed horizontally, there is no initial vertical velocity (V₀y = 0). The vertical displacement (Δy) is given as 1.9 cm, which we need to convert to meters (0.019 m). The acceleration due to gravity (g) is approximately 9.8 m/s².

Plugging in the values we know:

0.019 = (1/2) * 9.8 * t²

Simplifying the equation:

0.019 = 4.9 * t²

Rearranging to solve for t²:

t² = 0.019 / 4.9

t² ≈ 0.003878

Taking the square root of both sides to find t:

t ≈ √0.003878

t ≈ 0.06228 s

Therefore, the bullet's time of flight is approximately 0.06228 seconds.

(b) Finding the muzzle velocity:
To determine the muzzle velocity (V₀x), we can use the equation:

Δx = V₀x * t + (1/2) * a * t²

Since the bullet was fired horizontally, there is no acceleration in the horizontal direction (a = 0). The horizontal displacement (Δx) is given as 30 m.

Plugging in the values we know:

30 = V₀x * t

Substituting the value of t we previously found (t ≈ 0.06228 s):

30 = V₀x * 0.06228

Rearranging to solve for V₀x:

V₀x = 30 / 0.06228

V₀x ≈ 481.71 m/s

Therefore, the muzzle velocity of the bullet is approximately 481.71 m/s.