A bullet with a muzzle velocity of 200 m/s is fired horizontally from a rifle mounted on top of
a window ledge 7.8 m from the ground.
a.) How long will it take to hit the ground?
b.) What is the range of the bullet?
a)7.8 = 1/2 (9.8) t^2
b) R =vt
To find the time it takes for the bullet to hit the ground, we can use the equation for vertical motion:
h = (1/2)gt^2
Where:
h = height (7.8 m)
g = acceleration due to gravity (-9.8 m/s^2, assuming downward direction)
t = time
a.) To find the time it takes to hit the ground, we will solve the equation for t:
7.8 = (1/2)(-9.8)t^2
Multiply both sides by 2:
15.6 = -9.8t^2
Divide both sides by -9.8:
t^2 = 15.6 / -9.8
t^2 ≈ -1.59
Since the time cannot be negative, we can conclude that there is no real solution for t^2. This suggests that the bullet does not hit the ground.
b.) Since there is no time for the bullet to hit the ground, we cannot calculate the range of the bullet.
To solve this problem, we can use the equations of projectile motion. Let's break it down into two parts:
a.) How long will it take to hit the ground?
First, we need to find the time it takes for the bullet to reach the ground. We can use the equation:
h = (1/2) * g * t^2
where h is the vertical distance (7.8 m), g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time it takes to hit the ground.
Rearranging the equation, we get:
t^2 = (2h) / g
t = sqrt((2h) / g)
Substituting the given values, we have:
t = sqrt((2 * 7.8) / 9.8)
t ≈ 1.4 seconds
Therefore, it will take approximately 1.4 seconds for the bullet to hit the ground.
b.) What is the range of the bullet?
To calculate the range, we need to determine the horizontal distance covered by the bullet. In this case, since the bullet is fired horizontally, there is no initial vertical velocity (Vy = 0).
The equation to find the range (R) is:
R = Vx * t
where Vx is the horizontal component of the bullet's initial velocity, and t is the time it takes to hit the ground (found in part a).
Since the bullet is fired horizontally, the initial horizontal velocity is the same as the muzzle velocity (Vx = 200 m/s).
Substituting the given values:
R = (200 m/s) * 1.4 s
R ≈ 280 meters
Therefore, the range of the bullet is approximately 280 meters.