A basketball was kicked 20.0 m above horizontal with an initial velocity of 120.0 m/s at an angle of 35.0˚ above the horizontal.
(a) How long is the rocket in air before falling?
(b) What is the horizontal range for the rocket?
(c) With what speed does the rocket hit the ground?
Vo = 120m/s[35o].
Xo = 120*Cos35 = 98.3 m/s.
Yo = 120*sin35 = 68.8 m/s.
a. Y = Yo + g*Tr = 0.
68.8 - 9.8Tr = 0, Tr = 7.02 s. = Rise time.
h = ho + Yo*Tr + 0.5g*Tr^2 =
20 + 68.8*7.0 - 4.9*7.0^2 = 261.5 m. above gnd.
0.5g*Tf^2 = 261.5.
4.9*Tf^2 = 261.5, Tf = 7.31 s. = Fall time.
Tr+Tf = 7.02 + 7.31 = 14.33 s. = Time in air.
b. Range = Vo^2*sin(2A)/g = 120^2*sin(70)/9.8 = 1381 m.
c. Y = Yo + g*Tf = 0 + 9.8*7.31 = 71.6 m/s. = Ver. component.
V = Sqrt(Xo^2+Y^2) = Sqrt(98.3^2+71.6^2) =
To solve these problems, we can use the equations of motion for projectile motion. Here's how we can find the answers to each question:
(a) How long is the rocket in the air before falling?
In projectile motion, the vertical motion can be treated as free fall, so we can use the following equation:
h = viy * t - (1/2) * g * t^2
Where:
- h is the vertical displacement (20.0 m in this case)
- viy is the initial vertical velocity (vi * sin(θ))
- t is the time in the air
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Rearranging the equation, we can solve for t:
(1/2) * g * t^2 - viy * t + h = 0
Substituting the values:
(1/2) * 9.8 * t^2 - 120 * sin(35°) * t + 20 = 0
We can solve this quadratic equation to find the time t.
(b) What is the horizontal range for the rocket?
The horizontal range can be calculated using the equation:
range = vix * t
Where:
- range is the horizontal distance traveled by the projectile
- vix is the initial horizontal velocity (vi * cos(θ))
- t is the time in the air (which we calculated in part (a))
Substituting the values:
range = 120 * cos(35°) * t
(c) With what speed does the rocket hit the ground?
When the rocket hits the ground, its final vertical velocity (vf) is given by the equation:
vf = viy - g * t
Substituting the values:
vf = 120 * sin(35°) - 9.8 * t
The speed at which the rocket hits the ground can be calculated using the equation:
speed = sqrt(vfx^2 + vf^2)
Where:
- speed is the magnitude of the final velocity vector
- vfx is the final horizontal velocity (which remains constant throughout the motion)
Substituting the values:
speed = sqrt(vix^2 + vf^2)