Question

Q1 A ball is thrown upwards with some initial speed. It goes up to a height of 19.6m and then returns. Find (a) The initial speed (b) The time taken in reaching the highest point (c) The velocity of the ball one second before and one second after it reaches the maximum height (d) The time taken by the ball to return to its original position​

Answer 1

(a) 19.6 ms⁻¹

(b) 2 s

(c) 9.8 ms⁻¹

(d) 4 s

Explanation:

Constant Acceleration Equations (SUVAT)

`\boxed{\begin{array}{c}\begin{aligned}v&=u+at\\\\s&=ut+(1)/(2)at^2\\\\ s&=\left((u+v)/(2)\right)t\\\\v^2&=u^2+2as\\\\s&=vt-(1)/(2)at^2\end{aligned}\end{array}} \quad \boxed{\begin{minipage}{4.6 cm}$s$ = displacement in m\\\\$u$ = initial velocity in ms$^(-1)$\\\\$v$ = final velocity in ms$^(-1)$\\\\$a$ = acceleration in ms$^(-2)$\\\\$t$ = time in s (seconds)\end{minipage}}`

When using SUVAT, assume the object is modeled as a particle and that acceleration is constant.

Acceleration due to gravity = 9.8 ms⁻².

Part (a)

When the ball reaches its maximum height, its velocity will momentarily be zero.

Given values (taking up as positive):

`s=19.6 \quad v=0 \quad a=-9.8`

`\begin{aligned}\textsf{Using} \quad v^2&=u^2+2as\\\\\textsf{Substitute the given values:}\\0^2&=u^2+2(-9.8)(19.6)\\0&=u^2-384.16\\u^2&=384.16\\u&=√(384.16)\\\implies u&=19.6\; \sf ms^(-1)\end{aligned}`

Therefore, the initial speed is 19.6 ms⁻¹.

Part (b)

Using the same values as for part (a):

`\begin{aligned}\textsf{Using} \quad s&=vt-(1)/(2)at^2\\\\\textsf{Substitute the given values:}\\19.6&=0(t)-(1)/(2)(-9.8)t^2\\19.6&=4.9t^2\\t^2&=(19.6)/(4.9)\\t^2&=4\\t&=√(4)\\\implies t&=2\; \sf s\end{aligned}`

Therefore, the time taken to reach the highest point is 2 seconds.

Part (c)

As the ball reaches its maximum height at 2 seconds, one second before this time is 1 s.

Given values (taking up as positive):

`u=19.6 \quad a=-9.8 \quad t=1`

`\begin{aligned}\textsf{Using} \quad v&=u+at\\\\\textsf{Substitute the given values:}\\v&=19.6+(-9.8)(1)\\v&=19.6-9.8\\\implies v&=9.8\; \sf ms^(-1)\end{aligned}`

The velocity of the ball one second before it reaches its maximum height is the same as the velocity one second after.

Proof

When the ball reaches its maximum height, its velocity is zero.

Therefore, the values for the downwards journey (from when it reaches its maximum height):

`u=0 \quad a=9.8 \quad t=1`

(acceleration is now positive as we are taking ↓ as positive).

`\begin{aligned}\textsf{Using} \quad v&=u+at\\\\\textsf{Substitute the given values:}\\v&=0+9.8(1)\\\implies v&=9.8\; \sf ms^(-1)\end{aligned}`

Therefore, the velocity of the ball one second before and one second after it reaches the maximum height is 9.8 ms⁻¹.

Part (d)

From part (a) we know that the time taken to reach the highest point is 2 seconds. Therefore, the time taken by the ball to travel from the highest point to its original position will also be 2 seconds.

Therefore, the total time taken by the ball to return to its original position after it is thrown upwards is 4 seconds.