A tiger leaps horizontally from a 7.5-m-high rock with a speed of 3.0 m/s. How far from the base of the rock will she land?
The correct answer and explanation is :
Let’s solve it step-by-step:
First, find the time it takes the tiger to fall from the rock.
The vertical motion is free fall under gravity. We use the kinematic equation:
[
y = \frac{1}{2}gt^2
]
where:
- ( y = 7.5 \, \text{m} ) (height of the rock),
- ( g = 9.8 \, \text{m/s}^2 ) (acceleration due to gravity),
- ( t ) is the time to fall.
Solving for ( t ):
[
t = \sqrt{\frac{2y}{g}}
]
[
t = \sqrt{\frac{2(7.5)}{9.8}}
]
[
t = \sqrt{\frac{15}{9.8}}
]
[
t = \sqrt{1.5306}
]
[
t \approx 1.24 \, \text{seconds}
]
Next, find the horizontal distance.
Since there’s no air resistance and no horizontal acceleration, the horizontal distance ( x ) is:
[
x = v_x \times t
]
where:
- ( v_x = 3.0 \, \text{m/s} ) (horizontal velocity).
Substituting:
[
x = 3.0 \times 1.24
]
[
x = 3.72 \, \text{meters}
]
✅ Final Answer: The tiger will land approximately 3.72 meters away from the base of the rock.
Detailed Explanation (300+ words):
This problem is a classic projectile motion question where an object is launched horizontally. The key feature of horizontal launches is that the initial vertical velocity is zero because the tiger leaps straight out, not upwards or downwards.
The vertical motion is independent of the horizontal motion. Vertically, the only force acting on the tiger is gravity, which causes it to accelerate downwards at ( 9.8 \, \text{m/s}^2 ). To find how long it takes the tiger to fall, we apply the formula for displacement under constant acceleration, ( y = \frac{1}{2}gt^2 ). We solve this equation for time, and by substituting the given height (7.5 meters), we find the tiger falls for about 1.24 seconds.
Horizontally, there’s no acceleration because we ignore air resistance. Thus, the horizontal velocity remains constant at ( 3.0 \, \text{m/s} ) throughout the leap. Since velocity is constant, we can use the simple relationship ( x = v_x \times t ) to find the horizontal distance traveled. Multiplying the tiger’s constant horizontal speed by the time it’s in the air gives the distance she covers horizontally before hitting the ground.
This separation of horizontal and vertical components makes projectile motion problems manageable: you treat them separately and combine the results at the end.
Thus, the tiger will land approximately 3.72 meters from the base of the rock.