A stone is projected vertically upwards from point A, which is 1.8 m above the base of the building with a speed of 15 m·s⁻¹ - NSC Physical Sciences - Question 3 - 2017 - Paper 1

To find the time taken to reach the maximum height, we can use the kinematic equation:

$v_f = v_i + at$

At maximum height, the final velocity ($v_f$) is 0 m/s. The initial velocity ($v_i$) is 15 m/s, and gravitational acceleration ($a$) is -9.8 m/s²:

Setting the equation to zero:

$0 = 15 - 9.8t$

Rearranging gives:

$9.8t = 15 \Rightarrow t = \frac{15}{9.8} \approx 1.53 \, s$

To calculate the speed of the stone when it hits the building, we can use the same kinematic equation:

$v_f = v_i + at$

Where:

• $v_i = 15 \, m/s$ (initial speed)
• $a = -9.8 \, m/s²$ (deceleration due to gravity)
• $t = 2.4 \, s$ (total time before hitting the roof)

Substituting the values:

$v_f = 15 - (9.8 \times 2.4)$

Calculating gives:

$v_f \approx 15 - 23.52 \approx -8.52 \, m/s$

Since the negative sign indicates direction, the speed at which the stone hits the roof is approximately 8.52 m/s.

To find the height of the building, we must first find the displacement (Y - X) from the initial position A to the roof at point X. Setting the equation for displacement:

$Y = v_i t + \frac{1}{2} a t^2$

Calculating with $v_i = 15 \, m/s$, $a = -9.8 \, m/s^2$, and $t = 2.4 \, s$ gives:

$Y = 15(2.4) + \frac{1}{2}(-9.8)(2.4^2)$

Which results in:

$Y = 36 - 28.224 \approx 7.776 \, m$

The height of the building above point A is then:

$Height_{building} = Y + height_{A} = 7.776 + 1.8 = 9.576 \, m \approx 9.58 \, m$

The velocity-time graph has distinct segments. During the first phase (0 to 1.53 s), the velocity decreases linearly from 15 m/s to 0 m/s, illustrating the ascent to the maximum height at Y. In the next segment (1.53 s to 2.4 s), the velocity decreases further due to gravity, reaching the final velocity of approximately -8.52 m/s at point X. The graph should clearly mark:

i) Initial velocity at point A (15 m/s)

ii) Time at Y (1.53 s)

iii) Final velocity at point X (-8.52 m/s).