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Question: A batsman deflects a ball by an angle of \({{45}^{0}}\) without changing its initial speed which is ...

A batsman deflects a ball by an angle of 450{{45}^{0}} without changing its initial speed which is equal to 54kmph. What is the impulse imparted to the ball ? (Mass of the ball is 0.15 kg)

Explanation

Solution

Firstly, the impulse is the difference between the final momentum and initial momentum of the ball. Next, it is given that the total angle deflected us 45 degrees. If we place it in the coordinate system, the initial and final direction of the ball makes 22.5 degrees with the axis. Hence, calculate the momentum easily.

Formula used:
impulse=Pf−Piimpulse={{P}_{f}}-{{P}_{i}}

Complete step by step answer:
let us write down the given terms and quantities,
θ=450 ⇒u=54kmph=15mps ⇒m=0.15kg \begin{aligned} & \theta ={{45}^{0}} \\\ & \Rightarrow u=54kmph=15mps \\\ & \Rightarrow m=0.15kg \\\ \end{aligned}
Let us calculate the initial momentum of the ball before the batman hit the ball,
The ball makes an angle 22.5 degrees with the central axis.
Pi=−mvcos⁡θ ⇒Pi=−0.15×15×0.9239 ⇒Pi=−2.08kgm−1 \begin{aligned} & {{P}_{i}}=-mv\cos \theta \\\ & \Rightarrow {{P}_{i}}=-0.15\times 15\times 0.9239 \\\ & \Rightarrow {{P}_{i}}=-2.08kg{{m}^{-1}} \\\ \end{aligned}
Similarly, the momentum of the ball after the batsman hit it will be,
Pf=mvcos⁡θ ⇒Pf=2.08kgm−1 \begin{aligned} & {{P}_{f}}=mv\cos \theta \\\ & \Rightarrow {{P}_{f}}=2.08kg{{m}^{-1}} \\\ \end{aligned}
Now, the impulse is the difference between the final momentum and the initial momentum,
impulse=2.08−(−2.08) ∴impulse=4.16kgm−1 \begin{aligned} & impulse=2.08-(-2.08) \\\ & \therefore impulse=4.16kg{{m}^{-1}} \\\ \end{aligned}
Hence, the impulse is calculating in this way.

Additional information:
Momentum is a word that we hear used colloquially in everyday life. We are often told that sports teams and political candidates have "a lot of momentum". In this context, the speaker usually means to imply that the team or candidate has had a lot of recent success and that it would be difficult for an opponent to change their trajectory. This is also the essence of the meaning in physics, though in physics we need to be much more precise.
Momentum is a measurement of mass in motion: how much mass is in how much motion. It is usually given the symbol p.

Note:
In the above question, if we consider the coordinate axis, while calculating the initial momentum, the ball is travelling in the x axis, which is why the momentum is taken negative. In the final momentum, the ball moves in the opposite direction after getting hit, so it is positive.