5. A truck is travelling at a constant velocity of 16.7 m/s. A pumpkin is shot vertically (at a 90o angle to the ground)at a velocity of 120. m/s from the bed of a this pickup truck. The bed of the pickup truck is one meter high (from the ground). Complete the following questions assuming that the air resistance is negligible.

a) Draw the initial velocity vector vi of the pumpkin with the vix and viy also shown.

b) Plot the x and y components of velocity (vx and vy ) versus time for the pumpkin.

c) What time will the pumpkin take to reach its maximum height?

d) What is the maximum height of the pumpkin's trajectory (from the ground) ?

e) How far does the truck travel before the pumpkin lands back in it's bed?

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a) To draw the initial velocity vector vi of the pumpkin, we need to break it down into its x and y components (vix and viy).

The given information states that the pumpkin is shot vertically (at a 90° angle to the ground) and has a velocity of 120 m/s. Since the vertical component of the velocity is responsible for the pumpkin's upward motion and the horizontal component is responsible for its motion across the ground, we can use trigonometry to find the components.

The horizontal component, vix, will be 0 m/s since the pumpkin is shot vertically.
The vertical component, viy, can be found using the equation:
viy = v * sin(θ)
where v is the magnitude of the velocity (120 m/s) and θ is the angle with the horizontal (90°).

Therefore, viy = 120 * sin(90°) = 120 m/s

So, the initial velocity vector vi of the pumpkin can be drawn as a vertical line pointing upwards, with viy = 120 m/s and vix = 0 m/s.

b) To plot the x and y components of velocity (vx and vy) versus time for the pumpkin, we can use the equations of motion.

Since the truck is moving with a constant velocity of 16.7 m/s horizontally and there is no horizontal force acting on the pumpkin, the x component of velocity (vx) will remain constant at vix = 0 m/s.

The y component of velocity (vy) will change due to the acceleration due to gravity (g) acting on the pumpkin. The equation for the vertical motion of an object under constant acceleration is:
vy = viy - g * t
where vy is the vertical component of velocity at time t, viy is the initial vertical component of velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.

To plot the components versus time, we can choose a range of time values and substitute them into the equations to find the corresponding velocities:

For example, let's choose a range of time from 0 to 5 seconds. We can then calculate the vy values for each time step, using the equation vy = viy - g * t.

c) To find the time the pumpkin takes to reach its maximum height, we need to consider the vertical motion of the pumpkin under gravity.

When the pumpkin reaches its maximum height, its vertical component of velocity (vy) will be zero. Using the equation of motion vy = viy - g * t, we can set vy to zero and solve for t:

0 = viy - g * t

Solving for t, we get:
t = viy / g

Substituting the given values, t = 120 m/s / 9.8 m/s^2 = 12.24 s

Therefore, the pumpkin takes approximately 12.24 seconds to reach its maximum height.

d) To find the maximum height of the pumpkin's trajectory from the ground, we can use the equation for vertical displacement:

Δy = viy * t + (1/2) * g * t^2

At the maximum height, the vertical displacement (Δy) will be equal to the height of the truck bed, which is 1 meter.

Substituting the known values, we get:
1 m = (120 m/s) * t + (1/2) * (9.8 m/s^2) * (t^2)

Simplifying the equation, we can solve for t using the quadratic formula:
0 = (1/2) * (9.8 m/s^2) * t^2 + (120 m/s) * t - 1 m

Solving for t using the quadratic formula, we find two solutions: t1 ≈ -2.64 s and t2 ≈ 0.11 s. Since time cannot be negative, we discard t1 and keep t2 (the positive value).

Therefore, the maximum height of the pumpkin's trajectory from the ground is approximately 0.11 meters.

e) To find how far the truck travels before the pumpkin lands back in its bed, we need to consider the horizontal motion of the pumpkin.

Since the truck is moving with a constant velocity of 16.7 m/s, the horizontal distance traveled by the truck can be found using the equation:

Δx = vx * t

Since the horizontal component of velocity (vx) remains constant at 0 m/s, and we found that the time taken for the pumpkin to reach its maximum height is approximately 12.24 seconds, we can calculate the horizontal distance:

Δx = 0 m/s * 12.24 s = 0 meters

Therefore, the truck does not travel any distance before the pumpkin lands back in its bed.