Question

The kinetic energy, E joules, of an object is given by the formula E = 1/2mv^2, where m kg is the mass of the object and v m/s is the velocity of the object. The mass of a trolley is 4 kg and its original velocity is positive. When the velocity of the trolley is 1 m/s more than twice its original velocity, its kinetic energy increases by 80 joules. Find the original velocity of the trolley.

Answer 1

Answer:

Start off by making an inequality using the given information: Ek < 600 J (Where J = kg*m2/s2)It is important that the symbol used is a less than sign (<), and not a less than/equal to sign, because the question wants kinetic energies LESS THAN 600 J, but NOT equal to 600 J.

Using the kinetic energy formula, we get: (1/2)*m*v2 < 600 kg*m2/s2

Since we know the value of m = 3kg, we can plug that in to get: (1/2)*(3kg)*v2 < 600 kg*m2/s2

Now, units like kilograms and metres are not very important in mathematics, but for science courses like chemistry and physics, it is important that you keep track of units and make sure that they are the same on both sides of whatever inequality or equation you are working with.

For this inequality, we see that there are kg units on either side, so we can cancel those out and simplify to get: (3/2)*v2 < 600 m2/s2

Now, we multiply both sides of the inequality by (2/3) to get: v2 < 400 m2/s2

Now, here is the tricky part: because the velocity is squared, that means that it does not matter if v is a positive or a negative number, any number times itself will be a positive number (For example: 1*1 = 1 > 0, (-1)*(-1) = 1 > 0 ). Since we want to wind up with a positive value of 400, we take the square root of both sides of the inequality: $\sqrt{v^2}<\sqrt{400}$ --> this gives us $\left|v\right|<\left|20\right|=20$20 (Check by seeing that 202 = 400), so se know that our upper limit is v < 20. (Can check by punching v = 21 into our original inequality: (1/2)*(3)*(21)2 = 661.5 > 600)

However, v can also be as low as -20 because (-20)2 = 400. Can it be lower than -20, though? NO, because (1/2)*(3)*(-21)2 = 661.5 > 600 So our range of values is: -20 < v < 20, remembering to only use less than signs.

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