Physics help please?

Question

Question: a cannon sends a projectile towards a target a distance 1350m away. The initial velocity makes an angle of 27 degrees with the horizontal. The target is hit.
The acceleration of gravity is 9.8m/s^2. What is the magnitude of the initial velocity?


I've attmpted this problem like 50 times, but I really don't see how I can get the initial velocity without being given one of the components or something...any ideas?

Answer 1

You have a system of equations.
Let vx be the initial x velocity,
vy be the initial y velocity,
|v| be the magnitude of the velocity, and
t be the time of flight.

You know that:
vy = vx * tan (27°)

|v|² = vx² + vy²
= vx² (1 + tan² (27°))
= vx² / cos² (27°)

We now have the magnitude and the y component of the initial velocity in terms of the x component. If we solve for the x component, we therefore can get the magnitude easily enough.

Taking the height above ground to be zero at the cannon and the target, our equation for height as a function of time when the projectile lands becomes
0 = vy*t - ½gt²
=> vy = ½gt
=> t = 2 / g * vx * tan (27°)
(The other root, t = 0, corresponds to the time the projectile was fired.)

Also, the x component of the projectile's velocity is constant, giving
vx * t = 1350
=> 2 / g * vx² * tan (27°) = 1350
=> vx² = 675 * g / tan (27°)
=> |v|² = 675 * g / tan (27°) / cos² (27°)
= 675 * g / (sin (27°) * cos (27°))

I will leave it to you to plug in the numbers and take the square root to get |v|.

Answer 2

Look at this solution iteratively I did step 1- 4 in general and then came back to step 1 and went though steps again.

1. Compute total travel time t
t= 2t1
t1 is the time required for the projectile to attain max height
t1=Vv/g
t= 2Vv/g

2 The horizontal component is then in terms of horizontal distance S total time t is
Vh= S/t=S/(2Vv/g)= Sg/(2Vv)

3. compute Vv= Vh tan(27)
Vv= [S g/(2Vv)] tan(27)
Vv^2= 0.5 [S g tan(27)]

4. V= sqrt(Vh^2 +Vv^2) since Vh= Vv/tan(27)
V= sqrt(Vv^2 [ctan(27)]^2 +Vv^2)
V=Sqrt(Vv^2(1+ ctan[(27)]^2) finally
V=sqrt( 0.5 [S g tan(27)] [1+[ctan(27)]^2])

Just plug the numbers in