for the cannon ball one
I can use the equation of motion to determine the time taken by the cannonball to hit the ground:
h(t) = -0.5gt^2 + V₀t + h₀
where:
- g is the acceleration due to gravity, -9.8 m/s^2
- V₀ is the initial velocity, 75 m/s
- h₀ is the initial height, 50 m
I'll set h(t) to 0, since the ball will hit the ground when its height is 0, and solve for t:
0 = -0.5(-9.8)t^2 + 75t + 50
0 = 4.9t^2 + 75t + 50
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
where a = 4.9, b = 75, and c = 50.
t = (-75 ± √(75^2 - 4(4.9)(50))) / 2(4.9)
t = (-75 ± √(5625 - 980)) / 9.8
t = (-75 ± √4645) / 9.8
t ≈ 7.55 seconds (rounding to 2 decimal places)
Therefore, the cannonball will hit the ground after approximately 7.55 seconds.
for the second one
since I don't have any specific values for g, v₀, or h, I can still write the general form of the projectile motion equation:
h(t) = -0.5gt^2 + v₀t + h₀
where g is the acceleration due to gravity, v₀ is the initial velocity, and h₀ is the initial height.
I can then plug in these variable values into Desmos to plot the trajectory of the projectile.