From the edge of a cliff, a gun fires a pellet straight upward. The pellet’s speed after leaving the gun is 30 m/s; the cliff is 75 meters tall. Assume that the ground is at height = 0 and that the positive direction is upward. At what time does the pellet hit the ground?
The correct answer and explanation is :
The pellet will hit the ground 8.02 seconds after being fired.
Explanation:
This is a kinematics problem where we analyze the motion of a pellet fired straight up from a cliff. The pellet follows a parabolic trajectory under the influence of gravity.
Step 1: Identify Given Information
- Initial velocity: ( v_0 = 30 ) m/s
- Height of the cliff: ( h = 75 ) m
- Acceleration due to gravity: ( g = 9.81 ) m/s²
Step 2: Use the Kinematic Equation
The general equation of motion in free fall is:
[
y = h + v_0 t – \frac{1}{2} g t^2
]
Since we need to find when the pellet hits the ground, we set ( y = 0 ):
[
0 = 75 + 30t – \frac{1}{2} (9.81) t^2
]
Rewriting:
[
4.905t^2 – 30t – 75 = 0
]
Step 3: Solve the Quadratic Equation
Using the quadratic formula:
[
t = \frac{-(-30) \pm \sqrt{(-30)^2 – 4(4.905)(-75)}}{2(4.905)}
]
Solving this gives two values for ( t ):
[
t_1 = -3.06 \quad \text{(negative, not valid)}
]
[
t_2 = 8.02 \quad \text{(valid, as time cannot be negative)}
]
Thus, the pellet hits the ground after 8.02 seconds.
The plot above visually represents the pellet’s motion, showing that it first rises, reaches a peak, and then falls to the ground.