A bullet fired vertically upwards from a gun held 2m above the ground reaches its maximum height in 4sec, calculate its initial velocity and the total distance the bullet travels by the time it hits the ground
To solve this problem, we can use the equations of motion for an object in free fall.
Step 1: Determine the acceleration due to gravity
The acceleration due to gravity, denoted as "g," is approximately 9.8 m/s^2 on the surface of the Earth. We assume this value for our calculations.
Step 2: Calculate the final velocity at maximum height
When the bullet reaches its maximum height, its final velocity will be 0 m/s as it momentarily stops before falling back down. In this case, we can use the equation:
Final velocity (vf) = Initial velocity (vi) - (acceleration x time)
Since vf = 0, the equation becomes:
0 = vi - (g x 4)
Simplifying the equation:
vi = g x 4
Substituting g = 9.8 m/s^2:
vi = 9.8 m/s^2 x 4 = 39.2 m/s
Therefore, the initial velocity of the bullet is 39.2 m/s.
Step 3: Calculate the time taken to reach the ground
The total time for the bullet to reach the ground can be found by considering the upward and downward motion as two separate parts.
The upward motion takes 4 seconds, and the downward motion will also take 4 seconds. Therefore, the total time taken to reach the ground is 4 seconds + 4 seconds = 8 seconds.
Step 4: Calculate the total distance traveled
To find the total distance traveled by the bullet, we can use the equation:
Total distance = Initial velocity x total time
Substituting the values:
Total distance = 39.2 m/s x 8 s = 313.6 meters
Therefore, the bullet travels a total distance of 313.6 meters before hitting the ground.
To calculate the initial velocity of the bullet, we can use the equation of motion for vertical motion:
v = u + gt
Where:
- v is the final velocity (which will be zero when the bullet reaches its maximum height and when it hits the ground).
- u is the initial velocity.
- g is the acceleration due to gravity, which is approximately 9.8 m/s^2 (assuming the motion occurs near the surface of the Earth).
- t is the time taken.
Since the bullet reaches its maximum height in 4 seconds, we can substitute the values into the equation to solve for the initial velocity:
0 = u + (9.8 × 4)
0 = u + 39.2
Rearranging the equation to solve for the initial velocity (u):
u = -39.2 m/s
So the initial velocity of the bullet fired vertically upwards is -39.2 m/s. The negative sign indicates that the velocity is in the opposite direction to the motion (since the bullet is moving upwards).
To calculate the total distance traveled by the bullet before hitting the ground, we need to find the time it takes to reach the ground. Since the bullet travels upwards for 4 seconds, it will take the same amount of time to fall back down. Therefore, the total time in the air is 4 + 4 = 8 seconds.
Using the equation of motion for vertical motion:
s = ut + (1/2)gt^2
Where:
- s is the total distance traveled.
- u is the initial velocity.
- g is the acceleration due to gravity.
- t is the total time in the air (8 seconds).
Plugging in the values:
s = (-39.2 × 8) + (0.5 × 9.8 × 8^2)
s = -313.6 + 313.6
s = 0
Therefore, the total distance traveled by the bullet before hitting the ground is 0 meters. This result makes sense because the bullet returns to the same height at which it was fired, and the ground is at a distance of 2 meters below that height.
At the pinnacle, Vv=0
Vv=Viv+ gt
0=ViV-9.8*4 solve for initial velocity.
total distance?
H(4)=H(o)+Viv*4-9.8(4^2) solve for H(4)
total distance: 2*H(4)+2
Vv=
H(4)=H(0)+Viv*