Question

You toss a tennis ball straight upward. At the moment it leaves your hand it is at a height of 1.3 m above the ground, and it is moving at a speed of 8.8 m/s.(a) How much time does it ke for the tennis ball to reach its maximum height?(b) What is the maximum height above the ground that the tennis ball reaches?(c) When the tennis ball is at a height of 2.6 m above the ground, what is its speed?

Answer 1

Let's name all the variables that are given and that we know.

yi: initial height; yi = 1.3 m

vi: initial upward velocity; vi = 8.8 m/s

g: gravitational acceleration; g = 9.81 m/s^2

t: time; let's make this the independent variable.

y: height; let's make this a dependent variable.

v: velocity; let's make this another independent variable

We can derive equations relating time (t) in seconds and height (y) along with velocity (v).

y = yi + vi*t - (g/2)*t^2

v = vi - g*t

(This is done by integrating the acceleration function, but that's not important as long as you can recall this formula)

Now let's plug in the variables we know into both these equations.

y = 1.3 + 8.8t - (9.81/2)t^2 (let's call this the height function)

v = 8.8 - 9.81t (let's call this the velocity function)

Now we're ready to answer part (a).

From the start of the scenario, the ball continues to travel upward, until it reaches its maximum, at which point its velocity is momentarily zero, then starts to decrease.

This means that at its highest point, the ball has a velocity v = 0.

We can then use the velocity function and solve for t when the ball is at its highest.

0 = 8.8 - 9.81t

9.81t = 8.8

t = 0.897 seconds (answer to part (a))

Now we're ready to answer part (b).

Now that we know the time it takes for the ball to reach its maximum height, we can use that value for t (t = 0.897), and solve for y using the height function.

y = 1.3 + 8.8*0.897 - (9.81/2)*0.897^2

We can simply plug the right side into a calculator to solve for y.

y = 5.247 meters (answer to part (b))

Now we're ready to answer part (c).

So we're given an initial condition that y = 2.6m.

We can plug this value in for y in the height function and solve for t.

2.6 = 1.3 + 8.8t - (9.81/2)t^2

Solving using the quadratic formula (or any other method of your choice for solving quadratics), you actually get two answers. This is because the ball reaches 2.6m twice; first when it's going up, then again when it comes back down.

t = 0.162s, 1.632s

However, since the motion is parabolic, the velocity at 1.632s will just be negative of the velocity at 0.162s, so we can just plug in 0.162 for t, and use the velocity function to solve for v.

v = 8.8 - 9.81*0.162

Plugging the right side into a calculator,

v = 7.211 m/s (answer to part (c))