How Long to Reach Maximum Height in Projectile Motion?

[RuthlessTB]

Homework Statement

A ball is kicked with initial velocity 40 m/s at an angle of 37 degree above the horizontal. What is the time (in seconds) needed to reach the maximum height?

Homework Equations

Y-axis
h= Voy t - 0.5 g t^2
Vy= Voy - g t
Vy^2= Voy^2 - 2 g h

X-axis
d= Vox t
Vx= Vox

The Attempt at a Solution

I reached to a point where I couldn't figure out a way to continue..

I used this equation
h= Voy t - 0.5 g t^2
h= (40 sin(37)) t - 5 t^2

I have 2 unknowns..
I tried to use Vy=Voy - g t
to find the time, I got so far these 2 equations
h= (40 sin(37)) t - 5 t^2
0=(40 sin(37)) - 10 t

Is this right?

 

[lep11]

Homework Statement

I tried to use Vy=Voy - g t
to find the time, I got so far these 2 equations
h= (40 sin(37)) t - 5 t^2
0=(40 sin(37)) - 10 t
Is this right?

Yes, you can use either one to solve for t. If you use the upper formula, remember that you get the total flight time which you have to divide by 2 since we assume the projectile is symmetric.

 

[RuthlessTB]

Thank you, I used the second equation and the answer is 2.4 s.. is it right?

 

[lep11]

Thank you, I used the second equation and the answer is 2.4 s.. is it right?

Looks good.

 

[Nickie]

If yes, how do I continue?

I would say that your approach is correct so far. You have used the correct equations for projectile motion and set up the problem correctly. However, to solve for the time, you will need to use a system of equations. You have two equations with two unknowns (h and t), so you can solve for both of them simultaneously.

To do this, you can use substitution or elimination. In this case, substitution might be easier. You can rearrange your second equation to solve for t:

t = (40 sin(37))/10

Then, you can substitute this value for t into your first equation:

h = (40 sin(37)) * ((40 sin(37))/10) - 5 ((40 sin(37))/10)^2

Simplifying this equation will give you the maximum height in terms of the initial velocity and angle. From there, you can solve for t by plugging the values into your equation for time (t = (40 sin(37))/10).

Hope this helps!