Question

An object is launched at a velocity of 20 m/s with 25° upwards with the horizontal.a)What is the maximum height reached by the object at B? [3.64 m]b)What is the total flight time of the object from A-C? [1.72 s]c)What is the horizontal range of the object from A-C? [31.26 m]d)What is the magnitude of the velocity of the object just before it hits the ground before C? [20 m/s]

Answer 1

Answer:

The initial velocity of the object is: u=20m/su=20m/s

The angle with the horizontal is: θ=25∘θ=25∘

a)

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See full answer below.

Answer 2

In the problem V0 = 20 m/s, θ = 25° and g = 9.8 m/s2.

The height of the projectile is given by the component y, and it reaches its maximum value when the component Vy is equal to zero. That is when the projectile changes from moving upward to moving downward.(see figure above) and also the animation of the projectile.

Vy = V0 sin(θ) - g t = 0

solve for t

t = V0 sin(θ) / g = 20 sin(25°) / 9.8 = 0.86 seconds

Find the maximum height by substituting t by 0.86 seconds in the formula for y

maximum height y (0.86) = 20 sin(25°)(0.86) - (1/2) (9.8) (0.86) 2 = 3.64 meters

b) The time of flight is the interval of time between when projectile is launched: t1 and when the projectile touches the ground: t2. At t = t1 and t = t2, y = 0 (ground). Hence

V0 sin(θ) t - (1/2) g t2 = 0

Solve for t

t(V0 sin(θ) - (1/2) g t) = 0

two solutions

t = t1 = 0 and t = t2 = 2 V0 sin(θ) / g

Time of flight = t2 - t1 = 2 (20) sin(θ) / g = 1.72 seconds.

c) In part c) above we found the time of flight t2 = 2 V0 sin(θ) / g. The horizontal range is the horizontal distance given by x at t = t2.

range = x(t2) = V0 cos(θ) t2 = 2 V0 cos(θ) V0 sin(θ) / g = V02 sin(2θ) / g = 202 sin (2(25°)) / 9.8 = 31.26 meters

d) The object hits the ground at t = t2 = 2 V0 sin(θ) / g (found in part b above)

The components of the velocity at t are given by

Vx = V0 cos(θ) Vy = V0 sin(θ) - g t

The components of the velocity at t = 2 V0 sin(θ) / g are given by

Vx = V0 cos(θ) = 20 cos(25°) Vy = V0 sin(25°) - g (2 V0 sin(25°) / g) = - V0 sin(25°)

The magnitude V of the velocity is given by

V = √[ Vx2 + Vy2 ] = √[ (20 cos(25°))2 + (- V0 sin(25°))2 ] = V0 = 20 m/s