Question

A 0.500-kg potato is fired at an angle of 80.0° above the horizontal from a PVC pipe used as a "potato gun" and reaches a height of 110.0 m. (a) Neglecting air resistance, calculate the potato’s velocity when it leaves the gun. (b) The gun itself is a tube 0.450 m long. Calculate the average acceleration of the potato in the tube as it goes from zero to the velocity found in (a). (c) What is the average force on the potato in the gun? Express your answer in newtons and as a ratio to the weight of the potato.

Answer 1

(a)
`47.15ms^(-1)`

(b)
`2470.13ms^(-2)`

(c) 1235.06N and 252.05 as a ratio

Step-by-step explanation:

From Newton's second law of motion

F=ma where m is mass, a is acceleration and F is net force

Also from kinetic equation of motion, velocity and displacement are related using equation

`v^(2)=u^(2)+2sa`

Where v is final velocity, u is initial velocity, a is acceleration and s is displacement

From the free body diagram attached, final velocity at maximum height is 0 and initial velocity is
`usin80^(o)`

Also, the vertical component can be written as

`v_(y)^(2) }=u_(y)^(2) } -2gs` The negative sign before 2gs means displacement is opposite the gravitational force

Where g is acceleration due to gravity,
`u_(y)` is vertical component of initial velocity and
`v_(y)` is vertical component of final velocity

Since
`v_(y)` is 0

`u_(y)^(2) } =2gs`

s=110 and g is taken as 9.8

`u_(y)=√(2*9.8*110)=46.43275`

`u_(y)=46.43ms^(-1)`

Also, it's evident that the vertical component of initial velocity is
`u_(y)=u_(i)sin \theta` where
`\theta` is angle of projection and
`u_(i)` is resultant velocity

Making
`u_(i)` the subject we obtain
`u_(i)=\frac {u_(y)}{sin \theta}`

Since
`u_(y)` and
`\theta` are known as
`46.43ms^(-1)` and
`80^(o)` respectively, then
`u_(i)=\frac {46.43ms^(-1)}{sin 80^(o)}=47.15ms^(-1)`

Therefore, the velocity of potato is
`47.15ms^(-1)`

(b)

Displacement depends on length of tube hence s=0.450m hence going back to kinetic equation
`v^(2)=u^(2)+2sa`

The final velocity v is answer in part a which is
`47.15ms^(-1)`, initial velocity u is
`0ms^(-1)` hence the equation is re-written as

`v^(2)=2sa` and making a the subject we obtain

`a=\frac {v^(2)}{2s}`

`a=\frac {47.15^(2)}{2*0.450}=2470.13ms^(-2)`

Therefore, average acceleration is
`2470.13ms^(-2)`

(c)

From Newton's second law of motion, F=ma where m=0.500kg and a is
`2470.13ms^(-2)`

Therefore, the average force of potato is

F=0.5*2470.13=1235.06N

F=1235.06N

The weight, W of potato is given by W=mg

Taking R as ratio of average force and weight of potato

`R=\frac {F}{W}=\frac {F}{mg}` and since F=1235.06, m=0.500kg and g=9.8

`R=\frac {1235.06}{0.500*9.8}`=252.05

Therefore, ratio of average force to weight is 252.05