A cannonball is fired from ground level with an initial speed of 50\text{ m/s} at an angle… — JAMB UTME

A cannonball is fired from ground level with an initial speed of $50\text{ m/s}$ at an angle $\theta$ above the horizontal such that $\tan \theta = \frac{4}{3}$ . Assuming acceleration due to gravity $g = 10\text{ m/s}^2$ and neglecting air resistance, what is the magnitude of the velocity of the cannonball at the highest point of its trajectory?

$30\text{ m/s}$ Cevap
B
$0\text{ m/s}$
C
$40\text{ m/s}$
D
$50\text{ m/s}$

Cevap

30\text{ m/s}

At the maximum height of a projectile's flight, the vertical velocity component drops to zero due to gravity, while the horizontal component remains unchanged because horizontal acceleration is zero. For a launch speed of

50\text{ m/s}

at an angle with

\tan \theta = \frac{4}{3}

\cos \theta = \frac{3}{5}

. The magnitude of the velocity at apex is therefore equal to

u_x = 50 \times \frac{3}{5} = 30\text{ m/s}

Adım Adım Çözüm

Resolve the initial velocity vector into orthogonal horizontal and vertical components.

Given

\tan \theta = \frac{4}{3}

, the trigonometric ratios are

\cos \theta = \frac{3}{5}

and

\sin \theta = \frac{4}{5}

. Thus,

u_x = 50 \times \frac{3}{5} = 30\text{ m/s}

and

u_y = 50 \times \frac{4}{5} = 40\text{ m/s}

Resolving 2D projectile motion into independent orthogonal components simplifies kinematic analysis.

Determine the velocity components at the apex (highest point) of the trajectory.

At maximum height, the vertical velocity component

v_y = 0\text{ m/s}

. Since there is no horizontal acceleration, the horizontal velocity component remains

v_x = u_x = 30\text{ m/s}

Gravity acts purely vertically, reducing vertical velocity to zero at the apex while leaving horizontal velocity unchanged.

Calculate the magnitude of total velocity at maximum height.

v = \sqrt{v_x^2 + v_y^2} = \sqrt{30^2 + 0^2} = 30\text{ m/s}

The magnitude of a 2D velocity vector is determined by combining its orthogonal components using the Pythagorean formula.

Anahtar Kavram

Velocity components at maximum height in projectile motion

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