Question

A projectile is fired horizontally from a gun that is${}^{\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}}$ above flat ground, emerging from the gun with a speed of${}^{\mathbf{250}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}}$.(a)How long does the projectile remain in the air? (b)At what horizontal distance from the firing point does it strike the ground? (c)What is the magnitude of the vertical component of its velocity as it strikes the ground?

Short answer

(a) Time of flight of projectile is${}^{t=3.03s}$

(b) Horizontal distance from the firing point where it strikes the ground is${}^{x=758m}$

(c) Magnitude of the vertical velocity of the projectile as it strikes the ground is${}^{v_y=29.7m/s}$

Step-by-step solution

Given information

The initial velocity is in x direction, thus it is

$$\begin{gathered} v_{0x}=250m/s \\ h=45m \end{gathered}$$

To understand the concept of kinematic equations

This problem is based on kinematic equations that describe the motion of an object with constant acceleration. By applying second kinematic equation, time of the projectile can be found. Further, using the velocity of the projectile, the horizontal distance can be calculated. And the magnitude of the vertical velocity of the projectile as it strikes the ground can be computed using first kinematic equation.

Formula:

The displacement in kinematic equation is given by,

$d=v_{0}t+\frac{1}{2}a{t}^2$ (i)

The velocity in general is,

$v_0=\frac{d}{t}$ (ii)

$v=v_0+at$ (iii)

(a) To find the time of flight of projectile

It is given that,

$$\begin{gathered} \mathrm{d}-{\mathrm{v}}_0\mathrm{t}+\frac{1}{2}{\mathrm{at}}^2 \\ \mathrm{a}=\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2 \\ \mathrm{As}\mathrm{h}=45.0\mathrm{m} \\ {\mathrm{v}}_{0\mathrm{y}}=0\mathrm{m}/\mathrm{s} \\ \mathrm{t}=\sqrt{\frac{2\mathrm{h}}{\mathrm{g}}} \\ \mathrm{t}=\sqrt{\frac{2\left(45.0\mathrm{m}\right)}{9.8\mathrm{m}/{\mathrm{s}}^2}} \\ \mathrm{t}=3.03\mathrm{s} \end{gathered}$$

(b) To find the horizontal distance traveled by the projectile

It is given that,

$$\begin{gathered} {\mathrm{v}}_0=\frac{\mathrm{d}}{\mathrm{t}} \\ \mathrm{As}{\mathrm{v}}_{0\mathrm{x}}=250\mathrm{m}/\mathrm{s} \\ \mathrm{t}=3.03\mathrm{s} \\ {\mathrm{d}}_{\mathrm{x}}=\left(250\frac{\mathrm{m}}{\mathrm{s}}\right)\left(3.03\mathrm{s}\right) \\ {\mathrm{d}}_{\mathrm{x}}=758\mathrm{m} \end{gathered}$$

(c) To find the magnitude of the vertical velocity of the projectile as it strikes the ground

It is given that,

$$\begin{gathered} {\mathrm{v}}_{\mathrm{y}}={\mathrm{v}}_{0\mathrm{y}}+\mathrm{gt} \\ {\mathrm{v}}_{0\mathrm{y}}=0\mathrm{m}/\mathrm{s} \\ \mathrm{As},\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2 \\ \mathrm{t}=3.03\mathrm{s} \\ {\mathrm{v}}_{\mathrm{y}}=\left(9.8\frac{\mathrm{m}}{{\mathrm{s}}^2}\right)\left(3.03\right) \\ {\mathrm{v}}_{\mathrm{y}}=29.7\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, using the concept of free fall and kinematic equations, the time of flight and range of the projectile with given conditions can be found.