A ball P is dropped from rest from the top of a cliff of height 100\text{ m}. At the same … — JAMB UTME

A ball $P$ is dropped from rest from the top of a cliff of height $100\text{ m}$ . At the same instant, another ball $Q$ is projected vertically upwards from the base of the cliff along the same vertical line with an initial speed of $50\text{ m/s}$ . Taking $g = 10\text{ m/s}^2$ , at what height above the ground do the two balls meet?

A
$20\text{ m}$
B
$50\text{ m}$
$80\text{ m}$ Answer
D
$100\text{ m}$

Answer

The two balls meet at a height of

80\text{ m}

above the ground.

Equating the position equations of both objects gives

100 - 5t^2 = 50t - 5t^2

, which simplifies to

50t = 100

, so

t = 2\text{ s}

. Substituting

t = 2\text{ s}

into the vertical height formula gives

h = 80\text{ m}

above the ground.

Step-by-Step Solution

Set up equations of motion for both balls in terms of time

t

For ball

P

(dropped from top): height above ground

h_P = 100 - \frac{1}{2}gt^2 = 100 - 5t^2

.
For ball

Q

(projected from ground): height above ground

h_Q = ut - \frac{1}{2}gt^2 = 50t - 5t^2

Both balls move simultaneously under gravity along the same vertical line.

Equate the heights

h_P = h_Q

to find the time of meeting

t

100 - 5t^2 = 50t - 5t^2 \implies 50t = 100 \implies t = 2\text{ s}

The balls pass each other when their heights above the ground are equal.

Calculate the height above ground using

t = 2\text{ s}

h = 50(2) - 5(2)^2 = 100 - 20 = 80\text{ m}

Substituting

t = 2\text{ s}

into either height expression yields the position where they meet.

Key Concept

Relative vertical motion under gravity

Alternative Method

Using relative velocity: The relative acceleration between the two balls is

g - g = 0\text{ m/s}^2

. The relative speed of approach is constant at

50\text{ m/s}

. The initial separation is

100\text{ m}

, so time to meet is

t = \frac{100}{50} = 2\text{ s}

. Height above ground is then

h = 100 - \frac{1}{2}(10)(2)^2 = 80\text{ m}

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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