A helicopter is rising vertically at a uniform velocity of 14.7 m/s. When it is 196m from the ground, a ball is projected from it with a horizontal velocity of 8.5 m/s with respect to the helicopter. Calculate:
a.) when the ball will reach the ground
b.) where it will hit the ground
c.) what its velocity will be when it hits the ground
To solve this problem, we can separate the motion of the ball into horizontal and vertical components. Let's examine each part separately:
a.) When the ball will reach the ground:
To find the time it takes for the ball to reach the ground, we need to consider its vertical motion. The initial vertical velocity of the ball is zero since it is projected horizontally. The ball will accelerate downwards due to gravity, which we'll represent with the acceleration due to gravity, g = 9.8 m/s^2. The equation we can use to determine the time it takes for the ball to reach the ground is given by:
h = ut + (1/2)gt^2
Where:
h = vertical distance travelled (196 m)
u = initial vertical velocity (0 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation to solve for time (t):
196 = 0*t + (1/2)(-9.8)t^2
196 = -4.9t^2
Now, solving for t:
t^2 = 196 / (-4.9)
t^2 = -40
Since we cannot take the square root of a negative number in this context, there is no real solution for t. This means the ball will never reach the ground when it is horizontally projected from the helicopter with a constant horizontal velocity.
b.) Where it will hit the ground:
Based on our previous calculation, the ball will not hit the ground.
c.) What its velocity will be when it hits the ground:
Since the ball won't hit the ground, we cannot determine its velocity at impact.
To calculate when the ball will reach the ground, we need to determine the time it takes for the ball to fall from a height of 196 m. We can use the equation:
h = (1/2)gt^2
where:
h = height (in this case, 196m)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Rearranging the equation, we get:
t = sqrt(2h/g)
Plugging in the values, we get:
t = sqrt(2*196/9.8) = 8 seconds
Therefore, it will take the ball 8 seconds to reach the ground.
To determine where the ball will hit the ground, we need to calculate its horizontal distance traveled. Since the ball has a horizontal velocity of 8.5 m/s with respect to the helicopter, the horizontal distance it travels is given by:
d = v * t
where:
d = horizontal distance
v = horizontal velocity (8.5 m/s)
t = time (8 seconds)
Plugging in the values, we get:
d = 8.5 * 8 = 68 meters
Therefore, the ball will hit the ground 68 meters horizontally from the helicopter.
Lastly, to find the velocity of the ball when it hits the ground, we can break down its velocity into horizontal and vertical components.
The vertical velocity of the ball can be determined using the equation:
v = u + gt
where:
v = final vertical velocity (unknown)
u = initial vertical velocity (0 m/s, since it's projected horizontally)
g = acceleration due to gravity (approx. 9.8 m/s^2)
t = time (8 seconds)
Plugging in the values, we get:
v = 0 + 9.8 * 8 = 78.4 m/s
Therefore, the ball's vertical velocity when it hits the ground is 78.4 m/s.
The horizontal velocity of the ball remains constant at 8.5 m/s since there is no acceleration in that direction.
Combining the horizontal and vertical velocities, we can use the Pythagorean theorem to find the total velocity:
V = sqrt((v_h)^2 + (v_v)^2)
where:
V = total velocity (unknown)
v_h = horizontal velocity (8.5 m/s)
v_v = vertical velocity (78.4 m/s)
Plugging in the values, we get:
V = sqrt((8.5)^2 + (78.4)^2) ≈ 79.6 m/s
Therefore, the velocity of the ball when it hits the ground is approximately 79.6 m/s.
cheers
a. 7.99 s
b. 68.0 m
c. -63.7