Question

# A motorcycle and a bus are moving with the same momentum which of them has greater kinetic energy?

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Solution

## Step 1: Given data.It is given that, the momentum of motorcycle and bus is equal.Step 2: Formula to be used.The formula of momentum $p$ of an object having mass $m$ and velocity $v$ is defined as $p=mv$.The formula of the kinetic energy $K.E$ of an object having mass $m$ and velocity $v$ is defined as $K.E=\frac{1}{2}m{v}^{2}$.Step 3: Find the relation between the kinetic energy of the motorcar and the bus.Assume that, ${m}_{c}$ be the mass of the motorcycle and ${v}_{c}$ be the velocity of the motorcycle.Also, ${m}_{b}$ be the mass of the bus and ${v}_{b}$ be the velocity of the bus.We know that the mass of the bus is always greater than the mass of the motocycle.That is, ${m}_{c}<{m}_{b}$.Since, ${m}_{c}{v}_{c}={m}_{b}{v}_{b}...1$.which implies that, ${v}_{c}>{v}_{b}$.Divide both sides by $2$.$\frac{1}{2}{v}_{c}>\frac{1}{2}{v}_{b}$Now, multiply both sides by ${m}_{c}{v}_{c}$.$\frac{1}{2}{m}_{c}{v}_{c}^{2}>\frac{1}{2}{m}_{c}{v}_{c}{v}_{b}$From equation $1$.$\frac{1}{2}{m}_{c}{v}_{c}^{2}>\frac{1}{2}{m}_{c}{v}_{b}^{2}$This implies that the kinetic energy of the motorcycle is greater than the kinetic energy of the bus.Therefore, the motorcycle has greater kinetic energy.

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