A rifle is aimed horizontally at a target 52 m away. The bullet hits the target 2.2 cm below the aim point.
and what is the question?
You can figure out the muzzle velocity from those data. Divide the distance it travels by the time t that it takes to fall H = 2.2 cm.
t = sqrt(2H/g)
what is the speed?
To solve this problem, we can assume that the bullet follows a parabolic trajectory due to the effect of gravity. We need to find two key values: the initial velocity of the bullet and the time it takes for the bullet to travel 52 m horizontally.
Let's start by finding the initial vertical velocity component of the bullet. The vertical displacement of the bullet from the aim point is 2.2 cm, which is equivalent to 0.022 m. We can use the equation of motion for vertical displacement:
d = v_iy * t - (1/2) * g * t^2
where:
- d is the vertical displacement (0.022 m),
- v_iy is the initial vertical velocity component (to be found),
- t is the time of flight,
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the bullet is aiming horizontally, its initial vertical velocity component is 0 (v_iy = 0). With this information, the equation becomes:
0.022 = 0 * t - (1/2) * 9.8 * t^2
Simplifying the equation gives:
0.022 = -4.9 * t^2
Now we can solve for t:
t^2 = 0.022 / -4.9
t^2 ≈ -0.00449
Since time cannot be negative, we can conclude that there was an error in the calculation. Let's check the given values again and ensure their accuracy.
To find out the rifle's muzzle velocity, we can use the concept of projectile motion. Here's how we can approach this problem:
1. Identify the given information:
- The horizontal distance to the target (52 m).
- The vertical displacement from the aim point to the hit point (2.2 cm or 0.022 m).
2. Determine the horizontal component of the bullet's velocity:
Since the bullet is fired horizontally, its initial horizontal velocity remains constant throughout its flight. Therefore, the horizontal component of the bullet's velocity (Vx) is the same as the initial muzzle velocity (V₀).
3. Calculate the time of flight:
The time it takes for the bullet to travel horizontally to the target can be found using the formula:
Time (t) = Distance (d) / Velocity (V)
Since the horizontal distance is 52 m and the horizontal velocity (Vx) is the same as V₀, we can calculate the time of flight.
t = 52 m / V₀
4. Determine the bullet's vertical component of velocity:
To calculate the bullet's vertical component of velocity (Vy), we can use the vertical displacement (h) and the time of flight (t). The equation to find the vertical component is:
h = Vy * t + (0.5 * g * t^2)
Rearranging the equation to solve for Vy:
Vy = (h - 0.5 * g * t^2) / t
Where g is the acceleration due to gravity, approximately 9.8 m/s².
5. Calculate the vertical component of velocity and muzzle velocity:
Now that we know the bullet's vertical component of velocity (Vy), we can calculate the muzzle velocity (V₀) using the Pythagorean theorem:
V₀ = sqrt(Vx^2 + Vy^2)
Substitute Vx with V₀ and calculate the muzzle velocity.
By following these steps, you can find the muzzle velocity of the rifle. Remember to convert units consistently throughout the calculations.