Question #112488

A projectile fired at O follows a parabolic trajectory, given in parametric form by:

x = 86t and y = 96t – 4.9t2

where x and y are measured in meters and t in seconds. Determine:

(a) ax and ay at any time t

(b) vx and vy at any time t

(c) initial velocity, vo (velocity at pt O)

(d) velocity when t = 5s (hint: get first vx and vy at t = 5s)

(e) maximum height h

(f) horizontal distance L

x = 86t and y = 96t – 4.9t2

where x and y are measured in meters and t in seconds. Determine:

(a) ax and ay at any time t

(b) vx and vy at any time t

(c) initial velocity, vo (velocity at pt O)

(d) velocity when t = 5s (hint: get first vx and vy at t = 5s)

(e) maximum height h

(f) horizontal distance L

Expert's answer

**Solution.**

"x = 86t;" "y = 96t-4.9t^2;"

Since the trajectory of motion is a parabola, the motion of the projectile is described by the formulas:

"x = \\upsilon_0"_{x}"t;" "y = \\upsilon_0"_{y} "t + a_yt^2\/2;"

a) "a_x = 0; a_y = -9.8m\/s^2;"

b) "\\upsilon_x = 86m\/s ;""\\upsilon_y = 96-9.8t;"

c) "\\upsilon_0 =\\sqrt{\\upsilon_x^2 + \\upsilon_y^2 };" "\\upsilon_0 = \\sqrt{86^2 + 96^2 } = 128.9 m\/s;"

d) "\\upsilon_x = 86 m\/s; \\upsilon_y = 96 - 9.8\\sdot 5 = 47m\/s;"

e) "h_m = -\\upsilon_0"_{y}^{2} /2"\\sdot a_y;"

"h_m = -96^2\/2\\sdot(-9.8) = 470.2m;"

f) "L =\u03c5_0\n\u200b"_{x}"t_m;" "t_m =2\\sdot(-\\upsilon"_{0y}"\/a_y);" "t_m = 2\\sdot(-96\/-9.8) =19.6s;"

"L = 86\\sdot19.6 = 1684.9m;"

**Answer: **a) "a_x = 0; a_y = -9.8 m\/s^2;"

b) "\\upsilon_x = 86m\/s ;" "\\upsilon_y = 96-9.8t;"

c) "\\upsilon_0 = 128.9m\/s;"

d) "\\upsilon_x = 86 m\/s; \\upsilon_y = 47m\/s;"

e)"h_m = 470.2m;"

f)"L = 1684.9m;"

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