A ball is thrown vertically upward from the ground with an initial velocity of $39.2$ m/sec. If the only force considered is that attributed to the acceleration due to gravity, find:

(i) how long will it take for the ball to strike the ground?

(ii) the speed with which it will

Question

A ball is thrown vertically upward from the ground with an initial velocity of $39.2$ m/sec. If the only force considered is that attributed to the acceleration due to gravity, find:

(i) how long will it take for the ball to strike the ground?

(ii) the speed with which it will strike the ground?

(iii) how high the ball will rise?

Accepted Answer

> [!TLDR]
>
> With upward taken as positive, the ball's acceleration is $-g=-9.8	ext{ m/s}^2$ (gravity acts downward, opposing the upward throw). Integrate once to get velocity $v(t)$ (using the initial speed), integrate again to get position $s(t)$ (using $s(0)=0$ at the ground), then read off the three required quantities from $v(t)$ and $s(t)$.

**Step 1. Set up the acceleration equation.** Taking $g=9.8	ext{ m/s}^2$ and upward as positive,
$$a = \frac{dv}{dt} = -9.8.$$

**Step 2. Integrate to find $v(t)$.**
$$v = \int (-9.8)\,dt = -9.8t+c_1.$$
At $t=0$ (the instant of throwing), $v=39.2$, so $c_1=39.2$, giving
$$v(t) = 39.2-9.8t.$$

**Step 3. Integrate to find $s(t)$.**
$$s = \int (39.2-9.8t)\,dt = 39.2t-4.9t^2+c_2.$$
Measuring $s$ from the ground at $t=0$, $s(0)=0 \Rightarrow c_2=0$, so
$$s(t) = 39.2t-4.9t^2.$$

**Step 4. (i) Find when the ball strikes the ground.** The ball is back at the ground when $s=0$ (besides the start):
$$39.2t-4.9t^2=0 \Rightarrow t(39.2-4.9t)=0 \Rightarrow t=0 	ext{ or } t=\frac{39.2}{4.9}=8.$$
So the ball strikes the ground after $t=8$ seconds.

**Step 5. (ii) Find the speed on impact.** Substitute $t=8$ into $v(t)$:
$$v(8) = 39.2-9.8(8) = 39.2-78.4=-39.2 	ext{ m/s}.$$ …

Q.A ball is thrown vertically upward from the ground with an initial velocity of $39.2$ m/sec. If the only force considered is that attributed to the acceleration due to gravity, find:

Concept understanding — Properties of Integrals and Applications

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Q.A ball is thrown vertically upward from the ground with an initial velocity of $39.2$ m/sec. If the only force considered is that attributed to the acceleration due to gravity, find:

Unlock everything free for 14 days