Math Problem Statement

Math Problem Statement A cannonball is horizontally launch above a building with a height of 20m at a rate of 45m/s. (4pts) a. How long will it take the cannonball to reach the ground? b. What will be the range of the cannonball as it reaches the ground? C. What will be the magnitude of its velocity when it strikes the groud? D.at what direction will it strike the ground?

Solution

Problem Breakdown

We are given:

  • Initial horizontal velocity vx=45m/sv_x = 45 \, \text{m/s}
  • Height from which the cannonball is launched h=20mh = 20 \, \text{m}
  • The vertical motion is subject to gravity with g=9.8m/s2g = 9.8 \, \text{m/s}^2
  • Horizontal motion is constant because there’s no air resistance.

Let's solve the problem step-by-step:


Part A: Time to reach the ground

We know the vertical distance the cannonball falls is 20 meters. We can use the following kinematic equation for vertical motion:

h=12gt2h = \frac{1}{2} g t^2

Solving for tt, the time to reach the ground:

20=12(9.8)t220 = \frac{1}{2} (9.8) t^2

t2=204.9t^2 = \frac{20}{4.9}

t2=4.08t^2 = 4.08

t=4.082.02secondst = \sqrt{4.08} \approx 2.02 \, \text{seconds}

So, the cannonball will take 2.02 seconds to reach the ground.


Part B: Range of the cannonball

The horizontal distance (range) is calculated using the formula:

Range=vx×t\text{Range} = v_x \times t

Substitute the known values:

Range=45m/s×2.02seconds90.9meters\text{Range} = 45 \, \text{m/s} \times 2.02 \, \text{seconds} \approx 90.9 \, \text{meters}

So, the range of the cannonball is 90.9 meters.


Part C: Magnitude of velocity when it strikes the ground

The velocity when the cannonball strikes the ground is the resultant of its horizontal and vertical velocity components.

  • The horizontal velocity remains constant at vx=45m/sv_x = 45 \, \text{m/s}.
  • The vertical velocity vyv_y at the moment it strikes the ground is given by:

vy=g×t=9.8m/s2×2.02seconds19.8m/sv_y = g \times t = 9.8 \, \text{m/s}^2 \times 2.02 \, \text{seconds} \approx 19.8 \, \text{m/s}

Now, the total velocity vv is the resultant of vxv_x and vyv_y:

v=vx2+vy2v = \sqrt{v_x^2 + v_y^2}

v=452+19.82v = \sqrt{45^2 + 19.8^2}

v=2025+392.042417.0449.16m/sv = \sqrt{2025 + 392.04} \approx \sqrt{2417.04} \approx 49.16 \, \text{m/s}

So, the magnitude of the velocity when it strikes the ground is approximately 49.16 m/s.


Part D: Direction of the velocity when it strikes the ground

The direction (or angle θ\theta) at which the cannonball strikes the ground can be found using the following relation:

tan(θ)=vyvx\tan(\theta) = \frac{v_y}{v_x}

tan(θ)=19.845\tan(\theta) = \frac{19.8}{45}

θ=tan1(19.845)tan1(0.44)\theta = \tan^{-1}\left(\frac{19.8}{45}\right) \approx \tan^{-1}(0.44)

θ23.74\theta \approx 23.74^\circ

Thus, the cannonball strikes the ground at an angle of 23.74° below the horizontal.


Summary of Answers:

  • (a) Time to reach the ground: 2.02 seconds
  • (b) Range of the cannonball: 90.9 meters
  • (c) Magnitude of the velocity: 49.16 m/s
  • (d) Direction of the velocity: 23.74° below the horizontal

Do you want further details or have any questions?

Here are 5 related questions:

  1. What if the height of the building was doubled? How would the time change?
  2. How does air resistance affect the range and velocity?
  3. Can we calculate the maximum range of a cannonball given an initial angle?
  4. How do different gravitational accelerations affect the result (e.g., on the Moon)?
  5. What happens to the direction of the velocity if the initial horizontal speed is decreased?

Tip: When analyzing projectile motion, always treat horizontal and vertical motions separately.

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Math Problem Analysis

Mathematical Concepts

Kinematics
Projectile Motion
Vectors

Formulas

h = (1/2) g t^2
Range = v_x * t
v = sqrt(v_x^2 + v_y^2)
tan(θ) = v_y / v_x

Theorems

Pythagorean Theorem
Kinematic Equations

Suitable Grade Level

Grades 9-12