A railroad gun of mass M = 2.0 kg fires a shell of mass m=1.0 kg at an angle of θ= 40.0 ∘ with respect to the horizontal as measured relative to the gun. After the firing is complete, the final speed of the projectile relative to the gun (muzzle velocity) is v0=110.0 m/s . The gun recoils with speed vr and the instant the projectile leaves the gun, it makes an angle ϕ with respect to the ground.
What is vp, the speed of the projectile with respect to the ground (in m/s)?
vp=
What is ϕ, the angle that the projectile makes with the horizontal with respect to the ground (in degrees)?
ϕ=
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Its not helping, Can you please give out the formulas without the i' and j' etc??
Thanks
Vp = (M*V0*cos(theta))/(M + m) + V0*sin(theta)
Phi = tan_inverse((M+m)*tan(theta)/M)
To find the speed of the projectile (vp) with respect to the ground, we can use the concept of relative velocities. The velocity of the projectile relative to the ground is the vector sum of its velocity relative to the gun (v0) and the velocity of the gun (vr) relative to the ground.
vp = v0 + vr
To find the angle ϕ that the projectile makes with the horizontal with respect to the ground, we can use basic trigonometry. We know that the horizontal component of the velocity of the projectile (vp) is given by vp*cos(ϕ). Since the horizontal component of the velocity does not change upon firing, we can use the horizontal component of v0 (v0*cos(θ)) to determine ϕ.
ϕ = θ + arccos(v0*cos(θ)/vp)
Now, let's calculate the values. Since the values of θ, v0, and M are given, we need to find the value of vr first. To do that, we can use the principle of conservation of momentum.
Before the firing:
M*0 + m*0 = M*vr + m*v0
Simplifying the equation gives us:
vr = -(m*v0)/M
Now, we can substitute the values of v0, M, m, and θ into the equations to find vp and ϕ:
vr = -(1.0 kg * 110.0 m/s) / 2.0 kg, which equals -55.0 m/s
vp = 110.0 m/s + (-55.0 m/s), which equals 55.0 m/s
ϕ = 40.0° + arccos((110.0 m/s * cos(40.0°)) / 55.0 m/s)
Calculating this value gives us ϕ = 20.50°.
Therefore, vp = 55.0 m/s and ϕ = 20.50°.