Finding maximum height without time?

Question

A ball is thrown upward. Its initial vertical speed is 8.4 m/s, acceleration due to gravity is 9.8 m/s/s.
What is its maximum height reached? Answer in units of m.

Answer 1

You need to use the constant acceleration equation with t eliminated:

h = V²/2g

Answer 2

This can be solved in one of two ways:

1. Kinematics (Position, Velocity, Acceleration)
2. Conservation of Energy (Kinetic vs. Potential)

I think it is easier to solve as a conservation of energy problem, so that is what I'll show you. At the initial condition the ball is in your hand and has zero elevation (so zero potential energy) but it does have a velocity (so it does have kinetic energy). At the top of its flight it has attained some elevation, but for a split second it has stopped moving before it starts to fall again (zero kinetic energy).

Since energy cannot be either created or consumed, the kinetic energy at the beginning must all be converted to potential at the end. (Assuming that there is no loss of energy to air friction, which is a fine assumption for a slow moving ball... and there was no data about drag in the problem.)

KE = PE

See the links below to get some useful formulas for kinetic and potential energy.

m*g*y = 1/2*m*v²

You can divide both sides of the equation by the mass to remove that variable. Then just solve for y to get your height. If you do your math right you should get:

y = 3.6m

Answer 3

The maximum height of a projectile's flight can be calculated using the formula v2 = u2 + 2as and is again entirely reliant on the initial vertical velocity, as gravity would serve to apply a negative acceleration on the projectile. As before, v = 0 (that is, final velocity is zero at the maximum point of projection), u = uv, a = -9.8 m/s2
That is, 0 = uv2 - 19.6s
uv2 = 19.6s
s = uv2 / 19.6

Answer 4

Use the velocity equation:

v2 = v1 + at

You know v2 = 0, at the top. You're given v1, and you know a = -9.8. Solve for t, then use that in your equation for height:

y2 = y1 + v1*t + 1/2 * a * t^2

Answer 5

You don't need to know time. You know its initial velocity, its final velocity, and its acceleration. You have all the information you need to find the height it traveled. This is a simple plug and chug calculation.

Answer 6

Use conservation of energy

Initially there is no potential energy and finally, there is no kinetic energy

So 0.5*m*(8.4^2) = m*(9.8 m/s^2)*h

m's cancel out and solve for h

h = 3.6 m